STRUCTURAL DYNAMICS: RECENT ADVANCES
Proceedings of the 6th International Conference
Volume II
Proceedings of the Sixth International Conference on Recent Advances in
Structural Dynamics, held at the Institute of Sound and Vibration Research,
University of Southampton, England, from 14th to 17th July, 1997, co-sponsored by
the US Airforce European Office of Aerospace Research and Development and the
Wright Laboratories, Wright Patterson Air Force Base.
Edited by
N,S. FERGUSON
Institute of Sound and Vibration Research,
University of Southampton, Southampton, UK.
H.F. WOLFE
Wright Laboratory,
Wright Patterson Air Force Base, Ohio, USA.
and
C MEI
Department of Aerospace Engineering,
Old Dominion University, Norfolk, Virginia, USA.
© The Institute of Sound and Vibration Research, University of Southampton, UK.
ISBN no. 0-85432-6375
19970814 055
VV T.- , ..
PREFACE
The papers contained herein were presented at the Sixth International
Conference on Recent Advances in Structural Dynamics held at the Institute of
Sound and Vibration Research, University of Southampton, England in July 1997.
The conference was organised and sponsored by the Institute of Sound and
Vibration Research and co-sponsored by the Wright Laboratories, Wright Patterson
Air Force Base. We wish to also thank the following for their contribution to the
success of the conference: the United States Air Force European Office of Aerospace
Research and Development. The conference follows equally successful conferences
on the same topic held at Southampton in 1980, 1984,1988,1991 and 1994.
There are over one hundred papers written by authors from approximately
20 different countries, making it a truly international forum. Many authors have
attended more than one conference in the series whilst others attended for the first
time.
It is interesting to note the change in emphasis of the topics covered.
Analytical and numerical methods have featured strongly in all the conferences.
This time, system identification and power flow techniques are covered by even
more papers than previously. Also, there are many contributions in the field of
passive and active vibration control. Papers dealing with nonlinear aspects of
vibration continue to increase. These observations seem to reflect the trend in
current research in structural dynamics. We therefore hope that the present series
of International Conferences will play a part in disseminating knowledge in this
area.
We would like to thank the authors, paper reviewers and session chairmen
for the part they played in making it a successful conference.
My personal thanks go to the following individuals who willingly and
enthusiastically contributed to the organisation of the event:
Dr. H.F. Wolfe Wright Laboratories, WPAFB, USA
Dr. C. Mei Old Dominion University, USA
Mrs. M.Z. Strickland ISVR, University of Southampton, UK
Grateful thanks are also due to many other members of ISVR who contributed to
the success of the event.
N.S. Ferguson
Sixth International Conference on
Recent Advances in Structural Dynamics
Volume II
Contents
Page No.
INVITED PAPER
R.D. BLEVINS
On random vibration, probability and fatigue 881
ACOUSTIC FATIGUE I
58. J. LEE and K.R. WENTZ
Strain power spectra of a thermally buckled plate in
random vibration 903
59. S.A. RIZZI and T.L. TURNER
Enhanced capabilities of the NASA Langley thermal
acoustic fatigue apparatus 919
60. I. HOLEHOUSE
Sonic fatigue characteristics of high temperature materials
and structures for hypersonic flight vehicle applications 935
61 . M. FERMAN and H.F. WOLFE
Scaling concepts in random acoustic fatigue 953
ACOUSTIC FATIGUE II
62. H.F. WOLFE and R.G. WHITE
The development and evaluation of a new multimodal
acoustic fatigue damage model 969
63. B. BENCHEKCHOU and R.G. WHITE
Acoustic fatigue and damping technology in composite
materials 981
64. D. MILLAR
The behaviour of light weight honeycomb sandwich panels
under acoustic loading 995
65. P.D. GREEN and A. KILLEY
Time domain dynamic Finite Element modelling in acoustic
fatigue design 1007
SYSTEM IDENTIFICATION II
66.
U. PRELLS, A.W. LEES, M.I. FRISWELL and M.G. SMART
Robust subsystem estimation using ARMA-modelling in
the frequency domain
1027
67.
Y.Q. NI, J.M. KO and C.W. WONG
Mathematical hysteresis models and their application to
nonlinear isolation systems
1043
68.
M.G. SMART, M.I. FRISWELL, A.W. LEES and U. PRELLS
The identification of turbogenerator foundation models
from run-down data
1059
69.
C. OZTURK and A. BAHADIR
Shell mode noise in reciprocating refrigeration
compressors
1073
70.
T.H.T. CHAN, S.S. LAW and T.H. YUNG
A comparative study of moving force identification
1083
71.
P.A. ATKINS and J.R. WRIGHT
Estimating the behaviour of a nonlinear experimental multi
degree of freedom system using a force appropriation
approach
1099
POWER FLOW TECHNIQUES II
72.
R.S. LANGLEY, N.S. BARDELL and P.M. LOASBY
The optimal design of near-periodic structures to minimise
noise and vibration transmission
1113
73.
J.L. HORNER
Effects of geometric asymmetry on vibrational power
transmission in frameworks
1129
74.
M. IWANIEC and R. PANUSZKA
The influence of the dissipation layer on energy flow in
plate connections
1143
75.
H. DU and F.F. YAP
Variation analysis on coupling loss factor due to the third
coupled subsystem in Statistical Energy Analysis
1151
76.
S.J. WALSH and R.G. WHITE
The effect of curvature upon vibrational power
transmission in beams
1163
77.
S. CHOI, M.P. CASTANIER and C. PIERRE
A parameter-based statistical energy method for mid¬
frequency vibration transmission analysis
1179
PASSIVE AND ACTIVE CONTROL III
78. Y. LEI and L. CHEN
Research on control law of active suspension of seven
degree of freedom vehicle model 1195
79. M. AHMADIAN
Designing heavy truck suspensions for reduced road
damage 1203
80. A.M. SADRI, J.R. WRIGHT and A.S. CHERRY
Active vibration control of isotropic plates using
piezoelectric actuators 1217
81. S.M. KIM and M.J. BRENNAN
Active control of sound transmission into a rectangular
enclosure using both structural and acoustic actuators 1233
82. T.J. SUTTON, M.E. JOHNSON and S.J. ELLIOTT
A distributed actuator for the active control of sound
transmission through a partition 1247
83. J. RO, A. A-ALI and A. BAZ
Control of sound radiation from a fluid-loaded plate using
active constraining layer damping 1257
ANALYTICAL METHODS 11
84. E. MANOACH, G. DE PAZ, K. KOSTADINOV and
F. MONTOYA
Dynamic response of single-link flexible manipulators 1275
85. B. KANG and C.A. TAN
Wave reflection and transmission in an axially strained,
rotating Timoshenko shaft 1291
86. Y. YAMAN
Analytical modelling of coupled vibrations of elastically
supported channels 1329
87. R.S. LANGLEY
The response of two-dimensional periodic structures to
harmonic and impulsive point loading 1345
NONLINEAR VIBRATION III
88. H. OYANG, J.E. MOTTERSHEAD, M.P. CARTMELL and
M.L FRISWELL
Stick-slip motion of an elastic slider system on a vibrating
disc
1359
89.
R.Y.Y. LEE, Y. SHI and C. MEI
A Finite Element time domain multi-mode method for
large amplitude free vibration of composite plates
1375
90.
P. RIBEIRO and M. PETYT
Nonlinear forced vibration of beams by the hierarchical
Finite Element method
1393
91.
K.M. HSIAO and W.Y. LIN
Geometrically nonlinear dynamic analysis of 3-D beam
1409
92.
R.Y.Y. LEE, Y. SHI and C. MEI
Nonlinear response of composite plates to harmonic
excitation using the Finite Element time domain modal
method
1423
93.
C.W.S. TO and B. WANG
Geometrically nonlinear response analysis of laminated
composite plates and shells
1437
ANALYTICAL METHODS III
94.
R.S. HWANG, C.H.J. FOX and S. McWILLIAM
The free, in-plane vibration of circular rings with small
thickness variations
1457
95.
D.J. GORMAN
Free vibration analysis of transverse-shear deformable
rectangular plates resting on uniform lateral elastic edge
support
1471
96.
R.G. PARKER and C.D. MOTE, Jr.
Wave equation eigensolutions on asymmetric domains
1485
97.
A.V. PESTEREV
Substructuring for symmetric systems
1501
RANDOM VIBRATION I
98.
G.FUandJ. PENG
Anaytical approach for elastically supported cantilever
beam subjected to modulated filtered white noise
1517
99.
S.D. FASSOIS and K. DENOYER
Linear multi-stage synthesis of random vibration signals
from partial covariance information
1533
100.
CW.S.TOand Z. CHEN
First passage time of multi-degrees of freedom nonlinear
systems under narrow-band non-stationary random
excitations
1549
101.
C. FLORIS and M.C. SANDRELLI
Random response of Duffing oscillator excited by quadratic
polynomial of filtered Gaussian noise
1565
102.
S. McWILUAM
Extreme response analysis of non-linear systems to random
vibration
1581
103.
M. GHANBARI and J.F. DUNNE
On the use of Finite Element solutions of the FPK equation
for non-linear stochastic oscillator response
1597
RANDOM VIBRATION II
104.
T.L. PAEZ, S. TUCKER and C. O’GORMAN
Simulation of nonlinear random vibrations using artificial
neural networks
1613
105.
D.Z. LI and Z.C. FENG
Dynamic properties of pseudoelastic shape memory alloys
1629
106.
Z.W. ZHONG and C. MEI
Investigation of the reduction in thermal deflection and
random response of composite plates at elevated
temperatures using shape memory alloys
1641
SIGNAL PROCESSING I
107.
M. FELDMAN and S. BRAUN
Description of non-linear conservative SDOF systems
1657
108.
N.E. KING and K. WORDEN
A rational polynomial technique for calculating Hilbert
transforms
1669
109.
D.M. LOPES, J.K. FIAMMOND and P.R. WHITE
Fractional Fourier transforms and their interpretation
1685
SYSTEM IDENTIFICATION III
no. J. DICKEY, G. MAIDANIK and J.M. D’ARCHANGELO
Wave localization effects in dynamic systems 1701
111. P. YUAN, Z.F- WU and X.R. MA
Estimated mass and stiffness matrices of shear building
from modal test data
1713
112. YU. I. BOBROVNirSKn
The problem of expanding the vibration field from the
measurement surface to the body of an elastic structure 1719
113 M. AMABILI and A. FREGOLENT
Evaluation of the equivalent gear error by vibrations of a
spur gear pair
1733
ON RANDOM VIBRATION, PROBABILITY, AND FATIGUE
R. D. Blevins
Rohr Inc., Mail Stop 107X
850 Lagoon Drive
Chula Vista, California 91910
ABSTRACT
Analysis is made to determine the properties of a random process consisting of the
sum of a series of sine waves with deterministic amplitudes and independent, random
phase angles. The probability density of the series and its peaks are found for an arbitrary
number of terms. These probability distributions are non-Gaussian. The fatigue resulting
from the random vibration is found as a function of the peak-to-rms ratio.
1. INTRODUCTION
Vibration spectra of aircraft components often are dominated by a relatively small
number of nearly sinusoidal peaks as shown in Figure 1. The time history of this process,
shown in Figure 2, is irregular but bounded. The probability density of the time history,
shown in figure 3 only roughly approximates a Gaussian distribution and it does not exceed
2.5 standeird deviations.
The time history of displacement or stress of these processes over a flight or a take
off time can be expressed as a Fourier series of a finite number of terms over the finite
sampling period T.
N
y = 0,nCOs(u)-ntn + <^n), 0 < in < T, Un > 0 (l)
n=l
Each frequency Un is a positive, non-zero integer multiple of 27r/r. The following model is
used for the nature of the Fourier series: 1) the amplitudes a-n are positive and deterministic
in the sense that they do not vary much from sample to sample, 2) the phases (j>n are random
in the sense that they vary from sample to sample, they are equally likely to occur over
the range -oo < 0n < oo. This last condition implies that the terms on the right hand
side of equation (1) are statistically independent of each other.
We can generate an ensemble of values of the dependent variable Y by randomly
choosing M sets of N phase angles = 1,2..^), computing Y at some ffxed time
from equation (1), choosing another set of phases, computing a second value of Y and so
881
on until we have a statistically significant sample of M Y's. This random phase approach,
introduced by Rayleigh (1880), models a multi-frequency processes where each frequency
component is independent and whose power spectral density (PSD) is known.
The maximum possible (peak) value of equation (1) is the sum of the amplitude of
each term (recall > 0). The mean square of the sum of independent sine waves is the
sum of the mean squares of the terms.
N
^peak ~ 'y ^
n=l
= Na, for ai= 02= an = CL
(2a)
(2&)
X N N
Yrms = ^ f [Y anCO«(2wt„/T + <Pri)?dtn = 5
= |iVa^, for ai = 02 = an = a
The peak-to-rms ratio of the sum of N mutually independent sine waves thus is,
N N
I rms
1/2
n=l 71=1
= (2A/')^/^, for Oi = 02 = an = o.
(3a)
{Sb)
(4a)
(46)
Equation (4b) shows that the peak-to-rms ratio for an equal amplitude series increases from
2^/^ for a single term (N=l) and approaches infinity as the number of terms N approaches
infinity, as shown in Figure 4. The probability of Y is zero beyond the peak value. For
example, there is no chance that the sum of any four {N = 4) independent sinusoidal terms
will be greater than 8^/^ = 2.828 times the overall rms value.
2. PROBABILITY DENSITY OF A SINE WAVE
The probability density py (y) of the random variable Y is probability that the random
variable Y has values within the small range between y and y -b dy, divided by dy. p(Y)
has the units of 1/Y. Consider single a sine wave of amplitude a^, circular frequency uJri)
and phcLse
Y = On COs{0Jntn + 0 < < 277. (5)
Y is the dependent random variable. The independent random variables are tn or <l>n- The
probability density of a sine wave for equal likely phases p((l>n) = l/(27r), or equally likely
times, p[tn) = 1/T, is (Bennett, 1944; Rice 1944, art. 3.10),
wiy) =
77 ^(a^ - j/2) 1/2^ if <y <an\
0,
if I2/I > CLn
(6)
882
The probability density of the sine wave is symmetric about y = 0, i.e., pyiv) ~ PYi~y))
it is singular at y = Cn, and it falls to zero for jyl greater than an as shown in Figure 5.
The characteristic function of a random variable x is the expected value of
C{f) = r (7)
J —CO
and it is also the Fourier transform of the probability density function (Cramer, 1970,
pp. 24-35; Sveshnikov, 1965; with notation of Bendat, 1958). j = is the imaginary
constant. The characteristic function of the sine wave is found using equations (13) and
(14) and integrating over the range 0 < X < a^. (Gradshteyn, Ryzhik, Jeffrey, 1994, article
3.753).
Cn(f) = 2(7ra„)-‘ T" cos{27rfY))[l - {Y / dY = Jo(27r/a„), (8)
Jo
The characteristic function of a sine wave is a Bessel function of the first kind and zero
order (Rice, 1944, art. 3.16). Equations (6) and (8) are starting points for determining
the probability density of the Fourier series.
3. PROBABILITY DENSITY OF THE SUM OF N SINE WAVES
It is possible to generate an expression for the probability density of Fourier series
(equation l) with 1,2, 3, to any number of terms provided the sine wave terms are mutually
independent. This is done with characteristic functions. The characteristic function of the
sum of N mutually independent random variables (Y = Xi -j- X2 + -■■i- Xj\/) is the product
of their characteristic functions (Weiss, 1990, p.22; Sveshnikov, pp. 124-129),
c(/) = r .. r e^2-/{^.+^=+-
7 —00 7—00
N ^00 N
= n / = n CM)- (9)
The symbol 11 denotes product of terms. The characteristic function for the sum of N
independent sine waves is found from equations (8) and (9).
C{f)^
n!Li *^0 (27r/an) , unequal an
[Jo(27r/a)]^, ai == a2 = = a
(10)
The probability density of Y is the inverse Fourier transform of its characteristic function
(Sveshnikov, 1968, p. 129).
py{y) = r e-^^-fyc{f)df
7—00
(11)
883
By substituting equation (10) into equation (11) we obtain an integral equation for the
probability density of a N-term finite Fourier series of independent sine waves (Barakat,
1974).
.oo N
Pviy) = 2 / cos{2Tryf) { TT Jo(27r/an) } df,
''0 n=l
iV = 1,2,3...
(12)
If all N terms of the Fourier series have equal amplitudes a = ai — a2 = On = a/sr, then
this simplifies,
py(y) = 2 r cos(2iryf)[Jo{2wfa)fdf, N = 1,2,3... (13)
Jo
These distributions are symmetric about y — 0 as are all zero mean, sum-of-sine-wave
distributions. Figures 5 and 6 show results of numerically integrating equations (45) and
(46) over interval / = 0 to / = 15a using Mathematica (Wolfram, 1995).
Barakat (1974, also see Weiss, 1994, p. 25) found a Fourier series solution to equation
(45). He expanded the probability density of the N term sum in a Foui'ier series over the
finite interval -Ly < Y < Ly where Ly = ai + a2 + - + The result for unequal
amplitudes is ,
\v\<Ly.
i=L n=l
For equal amplitudes, ai = a2 = a,^ = o, Ly = Na, and
i=l
\y\ < Na.
(14)
(15)
Figure 6 shows that the Fourier series solution (equation 15) carried to 20 terms to be
virtually identical to numerical integration of equation (13) and it compares well with the
approximate solution. Note that theory requires py{\yT\ > Ly) = 0.
A power series solution for equation (13) can be found with a technique used by Rice
(1944, art. 16) for shot noise and by Cramer (1970) who called it an Edgeworth series. The
Bessel function term in equation (13) is expressed as an exponent of a logarithm which is
then expanded in a power series,
[Jo(27r/a)]-^ = ex'p{N ln[Jo[2'Kaf)\), (16)
= exvi-Nir'^a^f - (l/4)iV7r^o'‘/'' - (l/9)N-K^a^f + (n/192)Arx5a®/-)-
= + -1
Substituting this expansion into equation (13) and rearranging gives a series of integrals,
which are then solved (Gradshteyn, Ryzhik, Jeffrey, 1994, arts. 3.896, 3.952) to give a
884
power series for the probability density of the equal-amplitude N-term Fourier series sum.
VY{y) =
g^iFi[-2,V2,yV(2yA.,)l
-(
11
r(9/2)
(17)
192iV3 32iv2^ ^1/2
)^^iFi[-4,l/2,yV(2i;L.)l + -)^ \y\<Na
'PY{\y\ > = 0 and Yrms is given by equation (3b). There are two special func¬
tions in equation (17), the gamma function T and the confluent hypergeometric function
iFi[n,'y,z]. These are defined by Gradshteyn, Ryzhik, and Jeffrey (1994).
As N approaches infinity, the peak-to-rms (equation 4b) ratio approaches infinity, and
equation (51) approaches the normal distribution,
\im pY{y) = ~;^ — . (18)
N-*oo V^Yrms
as predicted by the central limit theorem (Cramer, 1970; Lin, 1976).
4. PROBABILITY DENSITY OF PEAKS
Theories for calculating the fatigue damage from a time history process generally
require knowledge of the peaks and troughs in the time history. This task is made simpler
if we assume that the time history is narrow band. If Y{t) is narrow band that is, that
each trajectory of Y{t) which crosses zero has only a single peak before crossing the cixis
again, then (1) the number of peaks equals the number of times the time history crosses
the axis with positive slope, and (2) only positive peaks occur for Y{t) >0 and they are
located at points of zero slope, dY{t)ldt = 0. Lin (1967, p. 304) gives expressions for the
expected number of zero crossings with positive slope (peaks above the axis) per unit time
for a general, not necessarily narrow band, process,
ElNo+]= f ypyy(0,y)<iy (19)
Jo
and the probability density of the peaks for a narrow band process.
= (20)
In order to apply these expressions, the joint probability distribution of Y and Y must be
established. The joint probability density function Pyriy^y) random variable
Y and Y is the probability that Y falls in the range between y and y + dy and y falls in the
range between y and y-\-dy, divided by dydy. The derivative of the sine wave Y (equation
12) with respect to time can be expressed in terms of Y ,
dY/dt = Y — -Gn^n sin{u)nt + 0n) == ~ Y^, \Y\ < On. (21)
885
The joint probability density is the inverse Fourier transform of its characteristic
function,
/CO poo
/ (30)
-oo J—oo
The proof of equations (28), (29), and (30) can be found in Chandrasekhar (1943), Willie
(1987), Weiss and Shmueli (1987), and Weiss (1994, pp. 21-26).
Since the probability is symmetric about y — y = 0, Pyriv^v) ~ Pyy(~S/j“y)) only
symmetric terms survive the integration. Substituting, equation (29) into equation (30)
and expanding gives and integral expression for the joint probability of y and Y.
Pyy(2/,y)== [ [ {JJ ^o(27ranY^/f + /|cj2)}cos(27r/iy)cos(27r/2y)d/i<i/2 (31)
It is also possible to expand the joint probability of Y and Y in as double finite Fourier
series. The result is:
.. oo oo N I . 7
Pyriv^y) = (^)2j}cos(i7ry/Ly)cos(/:7ry/L^)
aik = 1, i, /c> 0; 1/2, i — Qork = 0; 1/4, k = 0
(32)
(33)
The expected number of peaks per unit time and the probability distribution of the narrow
band peaks is obtained by substituting this equation into equations (19) and (20) and
integrating. The results are:
n—1
^0 k=0
TTUr
Ly ‘
(34)
y OO OO iV
:/-)' + (^P)}sin(i^A/Ly)
JuY J^Y
where
( 1/8, i = /c = 0,
_ J 1/4, i > 0, A: = 0
- 1 (l/2)[(-l)'‘- - ll/(fc7r)^ i = 0,fc > 0,
I [(-1)* - ll/(fe7r)2, i>0.fc>0.
(35)
(36)
If the frequencies are closely spaced so uJn^<^ and hence Ly » uLy , then one positive
peak is expected once per cycle,
£;[iVo+]=a;/(27r)
(37)
886
and the probability density of narrow band peaks becomes,
N
t=0 fc=0
71=1
(38)
Figure 7 shows probability density of narrow band peaks for N=2,3, and 4 equal amplitude
(tti = 1) equal frequency series using equation (38). Each sum in equation (38) was carried
to 40 terms.
A power series solution for equation (20) can be found if all N terms in the series have
equal amplitude and frequency. The result is
Pa(A) =
Y2
1 1
4iV 4ivy2
A^
32Nr,i,
•h...
In the limit as N becomes infinite these equations become,
(39)
pYi-(y>y) =
IttY Y
Zr/i j rms*^ rms
(40)
Pa{A) =
y2
^ rms
(41)
Equation (40) is in agreement with an expression given by Crandall and Mark (1963, p.
47) and equation (41) is the Rayleigh distribution.
Equations (20), (35), (38), and (41) are conservative when applied to non-narrow band
processes in the sense that any troughs above the axis (points with y > 0 and dYjdt = 0
but d^Yldt^ > 0) are counted as peaks (Lin, 1967, p. 304; Powell, 1958; Broch, 1963).
Equations (35), (38) and (41) can provide probability distributions for peaks of narrow
band processes as a function of the number of sine waves from one to infinity and thus
they model random processes with peak-to-rms ratios from 2^/^ to infinity.
5. FATIGUE UNDER RANDOM LOADING
Fatigue tests are most often made with constant-amplitude sinusoidal loading. The
number of cycles to failure is plotted versus the stress that produced failure and the data
is often fitted with an empirical expression. MIL-HDBK-5G (1994) uses the following
empirical expression to fit fatigue data,
log iVy = + B2log{S^ - S4), = 5(1 - R)^K (42)
Here Nf is the cycles to failure during sinusoidal loading that has maximum stress 5 per
cycle. R is the ratio of maximum to minimum stress during a cycle. R = — 1 is fully
887
reversed stress cycling. Bi though B4 are fitted parameters. With a little work, we can
put this expression in the form used by Crandall and Mark (1963, p. 113).
JV = cSJ*- (43)
where Sd = 5(1 - - B4, c = 10-®' , and b = -82- For cycling in a time history that
has non constant amplitude, Miner-Palmgren proposed that the accumulated damage is
the sum of the ratios of the number of cycles at each amplitude to the allowing number of
cycles to failure at that amplitude (equations 42 and 43).
D = ^«(Si)/lV^(S,) (44)
i
where n{Si) is the number of cycle accumulated at stress amplitude Si and Nj is the
number of stress cycles at this amplitude which would cause failure.
Following Miles(1954) and Crandall and Mark(1963), the expected fractional damage
for a random stress cycling in system with dominant cycling at frequency f in time t/, is
= (45)
where Pa{S) is the probability density of a stress cycle having amplitude S and Nf{S) is
the number of allowable cycles to failure at this stress. Failure under random loading is
expected when the expected damage is unity. Setting = 1 at time such that
ftd = Nd, the inverse of the expected number of random vibration cycles to failure is
This expression can be used to create a fatigue curve for random cycling given the proba¬
bility density of the random stress cycle amplitudes (p^(5)) and a fatigue curve (equation
42 with parameters Bi though B4 and R) for sinusoidal cycling.
Substituting the probability density expression for narrow band amplitude (equation
35) and for the fatigue curve(equations 42 or 43) into equation (46) and integrating, we
obtain an expression for the expected number of cycles to failure as a function of the
number of sine waves and their amplitudes. For N equal amplitude sine waves this is,
<"->■■ - I ^1"
(47)
V (2 + 6)L,(l-B)®= '
+
iirBi
(1 - R)^Ly
mBi
■2’
.1 3
1 T-irr-^ f-** ^
(1 -
iV(B^(l-B)®’-B4)2,
ALlil - B)2B3 '
i^7r^(£„(l - B)®° - Bif.
4L|(1 - H)2®»
888
Recall that for this case Ly = Na, the rms value is Y^ms — {l/2)Na and the peak-
to-rms ratio is Peak/Yrms = y/2N (equations 2 though 4). is the generalized
hypergeometric function which is a series of polynomials. It is described by Gradshteyn,
Ryzhik, and Jeffrey (1994).
It is also possible to establish the fatigue curve using the Rayleigh distribution (equa¬
tion 18) and the MIL-HDBK-5 fatigue curve (equation 42). The result is
(2^'^r[l + 5]((1 -
-BlF,[l +
^ 3 Bl
2’2’2y;2„,(i-ij)2B3«
(48)
Bl
rr2Y,^^{i-RYB,^
_orf3 + il + 3 Bl
4 2’ 2’ 2V;2„,(1 - ii)2B3
iFi[..] is the confluent hypergeometric function which is described by Gradshteyn, Ryzhik,
and Jeffrey (1994).
Much of the complexity of these last two equations arises from the term B4 which is
associated with an endurance limit in the fatigue equation. That is, equation (42) predicts
that sinusoidal stress cycling with stress less than 54/(1— R) ^=3 produces no fatigue damage.
If we set B4 = 0 to set the endurance limit to zero, then equation (48) simplifies to.
This result for cycles to failure under Gaussian loading without an endurance limit is also
given by Crandall and Mark (1963, p. 117).
Equations (47), (48) and (49) allow us to compute the fatigue curves of a material
under random loading from a fatigue curve generated under sinusoidal loading (equation
43) for narrow band random processes with any peak-to-rms ratio from 2^/^ to infinity.
6. APPLICATION
Figure 8 is the MIL-HDBK-5G fatigue curve for aluminum 2024-T3 with a notch
factor of Kt=4 under sinusoidal loading with various R values. The fitted curve shown in
the figure, gives the following parameters for equation (42).
Bl = 8.3, B2 = -3.30, Bz = 0.66, B4 = 8.4
889
The B2 and B3 are dimensionless. B4 has the units of ksi, that is thousands of psi, and
10^^ has units of These Bi,..B4 are substituted into equations (43), (47), (48),
and (49).
The fatigue curves under random loading are computed as follows, 1) the number of
sine waves N is chosen and this fixes the peak-to-rms ratio from equation (3b), 2) set of
values of rms stresses are chosen and for each the corresponding sine waves amplitudes are
computed using equation (3b), a = Srmsy/VN (note that the peak stress much exceed
S4=8.5 ksi), and 3)the cycles to failure are calculated from equation (47) for finite peak-
to-rms ratios and equation (48) for Gaussian loading (infinite peak-to-rms).
For single sine wave, the peak-to-rms ratio is 2^/^, equation 4b, and the fatigue curve
interms of rms stress is adapted from the empirical data fit (equations 42, 43) by substi¬
tuting 2^^‘^SrTns for the stress amplitude.
Nd = c(2^/25.n..(l - - B^r^ (50)
where b= -B2 and c = 10-®^ Some results are shown in Figure 9 for R=-l.
7. CONCLUSIONS
Analysis has been made to determine the properties of a random process consisting of
the sum of a series of sine waves with deterministic amplitudes and random phase angles.
The joint probability density of the sum and its first two derivatives is determined. The
probability density of the sum and narrow band peaks have been found for an arbitrary
number of statistically independent sine wave terms. The fatigue cycles-to-failure resulting
from these processes has been found.
1. The peak-to-rms ratio of the sum of mutually independent terms exceeds unity. If ail
terms have the same peak and rms values then the peak-to-rms ratio of the series sum
increases with the square root of the number of terms in the series. The probability
of the series sum is zero beyond a maximum value, equal to the sum of the series
amplitudes, and below the minimum value. Hence, he probability densities of the
finite series, their peaks, and their envelope are non Gaussian.
3. The formulas allow the direct calculation of the probability density of the series and its
peaks from its power spectra density (PSD) under the assumption that each spectral
component is statistically independent.
4. The fatigue curves of a material under random loading with any peak-to-rms ratio
from 2^/^ to infinity can be computed dfrectly from the fatigue curve of the material
under sinusoidal loading.
890
REFERENCES
Abramowitz, M. and LA. Stegun 1964 Handbook of Mathematical Functions, National
Bureau of Standards, U.S. Government Printing OfRce, Washington D.C. Reprinted by
Dover.
Bennett, W.R., 1944 Acoustical Society of America 15, 165. Response of a Linear Rectifier
to Signal and Noise.
Bendat, J.S., 1958 Principles and Applications of Random Noise Theory, Wiley, N.Y.
Chandrasekhar, S., 1943,Reweiys of Modem Physics, 15, 2-74. Also available in Wax, N.
(ed) Selected Papers on Noise and Stochastic Processes, Dover, N.Y., 1954.
Cramer, H., 1970 Random Variables and Probability Distributions, Cambridge at the Uni¬
versity Press.
Crandall, S.H., and C. H. Mark 1963 Random Vibrations in Mechanical Systems, Academic
Press, N.Y.
Department of Defense, 1994 Metallic Materials and Elements for Aerospace Vehicle Struc¬
tures, MIL-HDBK-5G.
Gradshteyn, I.S., I.M. Ryzhik, and A. Jeffrey 1994 Table of Integrals, Series, and Products
5th Ed., Academic Press, Boston.
Lin, P.K., 1976 Probabilistic Theory of Structural Dynamics, Krieger, reprint of 1967 edi¬
tion with corrections.
Mathematica, 1995 Ver 2.2, Wolfram Research, Champaign, Illiinois.
Miles, J., 1954 Journal of Aeronautical Sciences 21, 753-762. On Structural Fatigue under
Random Loading.
Powell, A., 1958 Journal of the Acoustical Society of America SO No. 12, 1130-1135. On
the Fatigue Failure of Structure due to Vibrations Excited by Random Pressure Fields.
Rayleigh, J.W.S. 1880 Philosophical Magazine X 73-78. On the Resultant of a Large
Number of Vibrations of the Same Pitch and Arbitrary Phase. Also see Theory of Sound,
Vol 10, art. 42a, reprinted 1945 by Dover, N.Y.. and Scientific Papers, Dover, N.Y., 1964,
Vol. I, pp. 491-496.
Rice, S.O., 1944 The Bell System Technical Journal 23 282-332. Continued in 1945 24 ,
46-156. Mathematical Analysis of Random Noise. Also available in Wax, N. (ed) Selected
Papers on Noise and Stochastic Processes, Dover, N.Y., 1954.
Shmulei, U. and G.H. Weiss 1990 Journal of the American Statistical Association 85 6-19.
Probabilistic Methods in Crystal Structure Analysis.
Sveshnikov, A. A, 1968 Problems in Probability Theory, Mathematical Statistics and Theory
of Random Functions Dover, N.Y., translation of 1965 edition, pp. 74, 116.
891
Tolstov, G.P., 1962 Fourier Series, Dover, N.Y., pp. 173-177. Reprint of 1962 edition.
Weiss, G.H., 1994 Aspects and Applications of the Random Walk, North-Holland, Amster¬
dam.
Weiss, S.H. and U. Shmulei, 1987 Physica 146A 641-649. Joint Densities for Random
Walks in the Plane.
Willie, L.T., 1987 Physica 141 A 509-523. Joint Distribution Function for position and
Rotation angle in Plane Random Walks.
Wirsching, RH., T.L. Paez, and K. Ortiz 1995 Random Vibrations, Theory and Practice,
Wiley-Interscience, N.Y., pp. 162-166.
892
A
an
Bn
C{f)
cih.h)
E[N,]
E[No^]
iFi
pEq
f
i
3
Jo
k
K
Ly
Ey
m
N
Nf
n
Bviy)
Py(^)
pxYi^,y)
S
t
T
Y
Y
X
aij
r
lij
(f>n
U
a>n
NOMENCLATURE
amplitude, peak, or envelope
amplitude of the nth sine wave, a^, > 0
fitted parameter in equation (42)
characteristic function with parameter /
joint characteristic function with parameters fi and /2
expected number of positive peaks per unit time
expected number of zero crossing with positive slope per unit time
confluent hypergeometric function (Gradshteyn, Ryzhik, Jeffrey, 1994, art. 9.210)
generalized hypergeometric function (Gradshteyn, Ryzhik, Jeffrey, 1994, art.
9.210)
parameter in Fourier transform
integer index
imaginary constant,
Bessel function of first kind and zero order
integer index
complete elliptic integral of first kind, equation (33a)
+ ^2 + •• + sum of amplitudes
ujiai + uj2a2 + - • + sum of velocity amplitudes
integer index
number of terms in series
cycles to failure
integer index, n=l,2,..N
cumulative probability, the integral of Py{x) from x=— co to y
probability density of random parameter Y evaluated at T = a:
joint probability density of X and Y evaluated at Y = y and X = x
stress
time, 0 < t < T
length of time interval
sum of N modes or terms, —Ly<Y < Ly
first derivative with respect to time of Y, —Ly <Y< Ly
a random variable
dimensionless coefficient, equation (33)
gamma function, r[(2n + l)/2] = 7r^/^2“’^(2n - 1)!!
dimensionless coefficient, equation (36)
Dirac delta function
3.1415926..
XiX2.-xi\i, product of terms
phase angle of the nth sine wave, a uniformly distributed independent random
variable
circular frequency, a positive (non zero) real number
circular frequency of the nth term, a non zero integer multiple of 27r/T
893
Figure 1 Spectrum of vibration of a component on a turbojet engine cowling. Note the
finite number of distinct peaks.
894
Figure 2 Sample of the time history associated with the spectrum of Figure 1. Note the
signal is bounded, irregular and quasi sinusoidal.
895
-4.0 0 0.0 2.0
MO, OF S.D.
Figure 3 Probability density of the time history of Figures 1 and 2. Note that the maximum
values do not exceed plus or minus 2.5 standard deviations.
Yrms * (Probability Density of Y)
-3-2-10 1 2 3
Y / Yrms
- Normal Distribution
- Sine Wave Distribution
o Equation . N=10
• Equation , N=1
Figure 5 Normal probability density (equation 18) and sine wave probability density (equa¬
tion 6) in comparison with results of numerical integration of equation (13) for N=1 and
N=10.
898
Yrms* (Probability Density of Y)
Yrms (Probability Density of A)
- Two Sine Waves
- Three Sine Waves
. Four Sine Waves
Figure 7 Probability density of peaks in narrow band series with equal amplitudes (ai =
02.. = 1) and frequencies.
900
10*
in' {o' 10* 10* 10*
FRTIGUE LIFE. CYCLES
FIGURE 3.2.3.1.8(h). Besi-fit SI N curves for noiched. K, ~ 4.0 of 2024-T3 aluminum alloy sheet,
longitudinal direction.
Figure 8 Fatigue curves for notched 2024-T3 aluminum alloy with Kt=4. MI1-HDBK-5G
(1994, p. 3-115)
902
ACOUSTIC FATIGUE I
Strain Power Spectra of a Thermally Buckled Plate
in Random Vibration
Jon Lee and Ken R. Wentz
Wright Laboratory (FIB)
Wright-Patterson AFB, OH 45433, USA
Abstract
Several years ago, Ng and Wentz reported strain power spectra measured
at the mid-point of a buckled aluminum plate which is randomly excited by an
electrodynamic shaker attached to the clamped-plate boundary fixture. We
attempt to explain the peculiar features in strain power spectra by generating
the corresponding power spectra by the numerical simulation of a single-mode
equation of motion. This is possible because the essential dynamics takes place
in the frequency range just around and below the primary resonance frequency.
1. Introduction
For high performance military aircraft and future high-speed civil transport
planes, certain structural skin components are subjected to very large acoustic
loads in an elevated thermal environment [1]. This is because high-speed
flights call for a very powerful propulsion system and thereby engendering
acoustic loads in the anticipated range of 135-175 dB. More importantly,
because of the aerodynamic heating in hypersonic flights and the modern trend
in integrating propulsion sub-systems into the overall vehicular configuration,
some structural components must operate at high temperatures reaching up to
1300°F. Hence, the dual effect of thermal and acoustic loading has given rise
to the so-called thermal-acoustic structural fatigue [2,3].
Generally, raising the plate temperature uniformly but with an immovable
edge boundary constraint would result in thermal buckling, just as one observes
flexural buckling as the inplane stress along plate edges is increased beyond a
certain critical value. This equivalence has been recognized [4,5] and
exploited in previous analytical and experimental investigations of the thermal-
acoustic structural fatigue [6,7,8]. An experimental facility for thermal-
acoustic fatigue, termed the Thermal Acoustic Fatigue Apparatus, was
constructed at the NASA Langley Research Center in the late 80’ s. Under the
acoustic loading of 140-160 dB, Ng and Clevenson [9] obtained some strain
measurements of root-mean-square value and power spectral density (PSD) on
an aluminum plate heated up to 250^. Later, Ng and Wentz [10] have
repeated the heated Aluminum plate experiment but by randomly exciting the
clamped-plate boundary fixture by a shaker, and thereby recovering similar
strain measurements.
It should be noted that Ng and his colleagues [7,9,10] were the first to
achieve sufficient plate heating to induce thermal buckling and thus observe the
erratic snap-through under the acoustic or shaker excitations. Here, by erratic
903
we mean that a snap-through from one static buckled position to another takes
place in an unpredictable fashion. We reserve the adjective chaotic for a snap-
through occurring under the deterministic single-frequency forcing [1 1,12]. It
has already been observed that certain of the buckled plate experiment can be
explained, at least qualitatively, by a single-mode model of plate equations.
This is also validated by a theoretical analysis. Indeed, we showed that a single¬
mode Fokker-Planck formulation can predict the high-temperature moment
behavior and displacement and strain histograms of thermally buckled plates,
metallic and composite [13,14].
In retrospect, a single-mode model has proven more useful than originally
intended. That is, the single-mode Fokker-Planck formulation of an isotropic
plate lends itself to predicting certain statistics of composite plates which are
simulated by multimode equations or tested experimentally by multimode
excitations. For a refined and more quantitative comparison, one must inject
more realism into dynamical models by including the multimode interactions.
However, before giving up the single-mode plate equation, there is an
important problem that this simple model is well suited for investigation. That
is, prediction of the strain PSD measurement by Ng and Wentz [10]. As we
shall see in Sec. 4, the strain PSD of a thermally buckled plate exhibits a strong
spectral energy transfer toward zero frequency, and thereby saturating
frequency range well below the primary resonance frequency. This downward
spectral energy transfer can be modeled quite adequately by the single-mode
plate equation without necessitating multimode interactions.
2. Equation of motion for the aluminum plate experiment
By the Galerkin procedure, the von Karman-Chu-Herrmann type of large-
deflection plate equations give rise to infinitely coupled modal equations [15].
However, much has been learned from a prototype single-mode equation for
displacement^ [13,14].
q + Pq + k„{l-s)q + aq^ = g„ + g{t), (1)
where the overhead dot denotes d/dt and the viscous damping coefficient is
P = 2^^ with damping ratio ^ . For the clamped plate, we have
;i„=f(r‘'+2rV3 + l),
s = rji + (1 -M) (1 + (r^+ ir^) /6] ,
a = ^{(7^+r'^+2^i) + |(i-/i^)[T(r^+r'^) +^(.r+r''T^
+ (r+47'‘)‘^ + (47+ 7"‘r^]} ,
&= (r‘'+2r^/3 +i)Sjj6.
Note that the expressions for s and g„ are specific to the typical temperature
904
variation and gradient profiles assumed in Ref. [15]. Here, 7 = b/a is the
aspect ratio of plate sides a and b, and fi is Poisson's ratio. The uniform plate
temperature is measured in units of the critical buckling temperature. The
maximum temperature variation on the mid-plate plane is denoted by and
TJ5g is the maximum magnitude of temperature gradient across the plate
thickness, where 5^ and 5 ^ are scale factors. Hence, 0 signifies no
temperature variation over the mid-plate plane, and 0 zero temperature
gradient across the plate thickness. Finally, g{t) denotes the external forcing.
The parameter s represents thermal expansion due to both the uniform
plate temperature rise above room temperature and temperature variation over
the mid-plate uniform temperature. The combined stiffness k^(l - s)q consists
of the structural stiffness k^q and thermal stiffness -sk^q , which cancel each
other due to the sign difference. It is positive for 5 <1, then Eq. (1) has the
form of Duffing oscillator with a cubic term multiplied by a , which represents
geometric nonlinearity of membrane stretching. For s >1 Eq. (1) reduces to
the so-called buckled-beam equation of Holmes [11] with a negative combined
stiffness. In contrast, denotes thermal moment induced by a temperature
gradient across the plate thickness; hence, it appears in the right-hand side of
Eq. (1) as an additional forcing. The interplay of the terms involving 5, a, and
g^ can best be illustrated by the potential energy [15]
U{q) = -go q + k^(X-s) (fn -H a . (2)
Fig. 1 shows that V{c^ is symmetric when g„ = 0. For s<l it has a single well
which splits into a double well as s exceeds unity. Note that the distance
between the twin wells increases as for large s (Fig. 1(b)). This
interpretation is valid approximately for go>^- That is, a positive g^ lowers
the positive side potential (^>0) and raises the negative side potential {q<0),
and thereby rendering the potential energy asymmetric.
U(q) ^(^1)
Fig. 1 Potential energy, (a) s<V, (b) ^ >1, w = ^k^(s - l)/a , d = - l))V4a.
( - ^.=0; --- 5„>0)
905
It must be pointed out that Eq. (1) is dimensionless and involves explicitly
only 7 and /i. For the aluminum plate experiment [10], 7=10 in./8 in. and
so that k^=Q3.9l and a =85. 33. If we further assume 5^- 0 for
simplicity, the thermal parameter reduces to s=T^. Previously, Eq. (1) was
used for the investigation of stationary Fokker-Planck distribution which
involves only the ratio p!F, where F is the constant power input [13,14].
Hence, nondimensionalization has indeed spared us from specifying in detail
other plate parameters. Things are however different in numerical simulation
because we must know the characteristic scales to correctly interpret time-
dependent solutions. By retracing the derivation, we find that the dimensionless
quantities in Eq. (1) are (Eq. (IV. 1) in Ref. [1])
q-qlh, t-t/t*, g = glg*, (3)
where the overhead bar denotes the physical quantity. Here, the plate thickness
h, t*={b/Kf.y[ph/D, and g^=p(h/t*)^ are the characteristic length, time, and
force, respectively (p = mass density, D=Eh^f\2{\~p}) , £= Young’s
modulus of elasticity), as listed in Table L We now rewrite Eq. (1) with the
numerical coefficients (Table I). _
q + 0.0978? + 23.91(1 -s)q + 85.33?^ = (4)
where g{t) has the unit of psi.
Table I. Parameter values for the aluminum plate experiment
7, «
10in./8in., VoX 0.01, 23.910, 85.332
5.. 5,
p
o
h, f*, g'^-
0.05 in., 3.305 lO'^sec.^, 5.806 1 O'’ psi^
(+) p = 0.098 Ib/in^ and E =1.03 10’ psi.
3. Monte-Carlo simulation
Because of 5 =0, Eq. (4) has the standard form of Duffing (s<l) and
Holmes (s>l) oscillators. In stead of a single frequency for forcing g{t) [11,
12], in Monte-Carlo simulation all forcing frequencies are introduced up to a
preassigned maximum so that forcing represents a plausible physical
realization. Of course, particular interest here is a constant PSD. We shall
begin with generation of a time-series for random processes with such a PSD.
3.1 Random forcing time-series
We adopt here the procedure for generating a time-series of Shinozuka
and Jan [16], which has been used for a oscillator study [17] and extensively
for structural simulation applications by Vaicaitis [18,19]. Since it relies
heavily on the discrete fast Fourier transforms, such as FFTCF and FFTCB
subroutines of the IMSL library, it is more expedient to describe the procedure
operationally rather than by presenting somewhat terse formulas. Let us
906
introduce Nj: frequency coordinates which are equally spaced in
the band width A/=/^3x/A^^. Now, the task is to generate a time-series of total
time T that can resolve up to . Assume T is also divided into time
coordinates with the equal time interval At=T/Nj, From the time-frequency
relation r=l/A/, we find ^At. If we choose
N
Nf = (5)
is the Nyquist frequency, consistent with our original definition of the
upper frequency limit of resolution.
A random time-series with a constant PSD can be generated in the
following roundabout way. We begin by assuming that we already have a
forcing power spectrum ^g{f) of constant magnitude over [0,/njax]- Such a
PSD may be represented by a complex array A„= VC exp(~27rz0„) (n = 1,
Nf), where takes a random value distributed uniformly in [0, 1]. Clearly the
magnitudes of are C, hence We then enlarge the complex array
A„ by padding with zeros for n =Ny+l, and Fourier transform it to
obtain a complex array B^{n= N^). The random time-series for is
now given by the real part of
= Real part of (n = 1, N^) (6)
As it turns out, when g „ is padded with zeros for the imaginary components
and Fourier transformed, we recover the original array A„ (w=l, Nf) with
Since the spectrum area is nothing but total forcing power
<g^> (say, in psi^), we can relate C with the variance <g^> of pressure
fluctuations, which is often expressed by the sound pressure level (SPL) in dB,
according to SPL=10 log<g2 >/p2, where p=2,9 10*^ psi. Hence,
c = - . (7)
/max
Here, Eqs. (6) and (7) defined heuristically are meant to explain the
corresponding formulas (2) and (12) in Ref [18].
For the numerical simulation we first note that the resonance frequency of
Eq. (1) is /^=-y^/27r« 0.778 for s ~ 0. This gives the dimensional resonance
frequency fjt*~235.5 Hz which is somewhat larger than the experimental
217.7 Hz (Fig. 3(a)). As shown in Table II, we assign (~9/r) because
the electrodynamic shaker used in the experiment [10] has the upper frequency
limit 2000 Hz.
Table II. Dimensionless parameter values for the numerical simulation
at, N, _ 7, 8192, 4096
At, T 0.071. 585
907
3.2 Displacement power spectrum
Under a random time integration of Eq. (4) yields a time-series for q^.
We first comment on the time integration. Although there are special solvers
[17,20] proposed for stochastic ordinary differential equations (ODEs), we
shall use here the Adams-Bashforth-Moulton scheme of Shampine and Gordon
[21], which has been implemented in DEABM subroutine of the SLATEK
library. Although DEABM has been developed for nonstochastic ODEs, its
use for the present stochastic problem may be justified in part by that one
recovers linearized frequency response functions by the numerical simulation
(Sec. 3.4). Obviously, this does not say anything about the strongly nonlinear
problem in hand, and it should be addressed as a separate issue. In any event,
DEABM requires the absolute and relative error tolerances, both of which are
set at no larger than 10“^ under the single-precision algorithm for time
integration. Note that actual integration time steps are chosen by the
subroutine itself, commensurate with the error tolerances requested. Recall
that is updated at every time interval Ar, and we linearly interpolate the
forcing value within A? .
We begin time integration of Eq. (4) from the initial configuration at the
bottom of the single-well potential, ^(0)=p(0)=0, for 5' <1 and the positive
side double-well potential, q{0)=^kj,s~\)/a andp(0)=0, for j >1. And we
continue the integration up to T. By Fourier transforming time-series q^, we
obtain displacement power spectrum 0^(/). This process of integrating and
transforming is repeated over three contiguous time ranges of 7, and the
successive PSDs are compared for stationarity. Since it is roughly stationary
after three repetitions, we report here only the PSD of the third repetition.
From the stationary input-output relation [22] where
is the magnitude of system frequency response function, we write
= (S)
Since O (/)=C, the and would have a similar functional
s
dependence upon /, Hence, we call them both the displacement PSD.
3.3 Strain power spectrum
Although displacement is the direct output of numerical simulation, one
measures strain rather than the displacement in plate experiment. At the
present level of plate equation formulation, the strain e is given
by the quadratic relation
e = + C^q + C2(f‘ ■> (9)
where C,- are given at the middle {x/a = y/b =1/2) of a clamped plate as follows
(Appendix D of Ref [13])
908
c =
(/+2yV3+l)?;g,
3(1+At)(y"+1)
{l-li}-
q =
8y^
3 ’
r 32 fy^ 5n (l-/iyV4)
^ 9 [2 16 2(y+y-‘f (y+4y-‘)^ (4y + y‘fJ
For we have C^=0, Q =4.17, and Q =2.77 (Table I). Hence, Eq. (9)
engenders only the linear and quadratic transformations, but no translation. In
any event, translation has no effect on the spectral energy contents. By Fourier
transforming time-series (n=l, A^^), we obtain strain power spectrum
. Although the forcing PSD is not constant, one computes the forcing
spectrum ratio as in Eq. (8) and call it the magnitude square of strain frequency
response function for the lack of a better terminology.
3.4 The linear oscillators
For the pre-buckled (5 <1) linear oscillator (a= 0) we rewrite Eq. (4) in
standard form _
q + + 0)1(1 -s)q = (10)
where col=k^, and obtain
I H^(f)^ = [(0)1(1 -s)- + (An^co^ffT'- (11)
As shown in Fig. 2(a), the numerical simulation of Eq. (10) recovers
as given by Eq. (11) over the entire frequency range. Although the simulation
of Fig. 2(a) was carried out with SPL=130 dB, it does not depend on SPL
since Eq. (10) is linear. Physically speaking, Eq. (10) oscillates in a single-well
potential (Fig. 1(a)). Since the potential energy has two wells (Fig. 1(b)) for
s >1, we linearize Eq. (1) around the positive side potential well by the
transformation q=q'+^k^{s-l)la . Hence, the corresponding linear oscillator
is
(a) (b)
Fig. 2 Linear frequency response functions, (a) Displacement; (b) Strain
(j = 0; - Numerical simulation; • Eq. (U))
909
q' + 2^0)^ q’ + Icolis -V)q' = (12)
s
In parallel to Eq. (1 1), the frequency response function of a post-buckled plate
I = [(^-colis - 1) - 4;rV^)2 + . (13)
The resonance frequency f=co^^2{s-l)/27J: of a post-buckled (s >1) plate
should be compared with f=co^^2{\~s)/2n of the pre-buckled (j <1) plate.
Now, for the linear oscillators we see that is also given by Eq. (1 1)
and (13) for ^ <1 and >1, respectively (Fig. 2(b)). This is because the spectral
energy distribution is not at ail affected by a linear transformation.
4. Displacement and strain power spectra
As we shall see in Sec 4.1, the experimental strain PSD exhibits downward
spectral energy transfer toward zero frequency, so that there is a considerable
spectral energy buildup below the resonance frequence as SPL is raised.
Moreover, it also involves an upward spectral energy transfer which then
contributes to both the increased resonance frequency and broadened
resonance frequency peak. Since spectral energy transfers take place around
and below the primary resonance frequency, it is possible to depict the
downward and upward spectral energy transfers by the numerical simulation of
Eq. (4) without necessitating multimode interactions. We shall first discuss the
characteristic features of experimental strain PSDs.
4. 1 Experimental strain PSD
Of the spectra reported in Ref. [10], we consider the following two sets.
One is the nonthermal set (^=0) consisting of two PSDs of small and large
SPLs. The other is the post-buckled set (5=1.7) of four PSDs. For the
convenience of readers, we have reproduced in Figs. 3 and 4 the selected PSDs
from Ref. [10] by limiting the upper frequency to 600 Hz, and the pertinent
data are summarized in Table m.
Table in. Strain power spectra of experiment and numerical simulation
Fig. 4(a) Fig. 6a
Fig. 4(b) Fig. 6b
Fig. 4(c) Fig. 6c
Fig. 4(d) Fig. 6d (*)
Fig. 7
Fig. 8
Fig. 9
Fig. 10
(*) Computed from the acceleration a measured in units of g.
910
Fig. 4 Experimental strain PSD =1.7). (a) 130.1dB; (b) 142dB; (c) 151.5dB; (d) 154.6dB
The following observations are drawn from the experimental PSDs. First,
for the nonthermal plate
Figure 3 fa): Compare the measured strain fr-2\l Hz with the theoretical
displacement 235 Hz of Eq. (4). Note that a small spectral energy peak is
found at 467 Hz which is about twice (-2.15) the strain value.
911
Figure Sfb): With SPL~150 dB the strain increases to 240 Hz and the
spectral width at the half resonance peak has nearly doubled. The spectral
energy buildups at zero and 515 Hz are more noticeable than in Fig. 3(a).
Again, 515 Hz is about twice (-2.15) the primary strain f,.
(a) (b)
f f
Fig. 5 Numerical simulation results under .y=0 and SPL=130 dB.
(a) Displacement ( - simulation,* Eq. (11)); (b) Strain ( - simulation, • Eq. (11));
(c) PSD averaged over 12 frequency intervals ( - displacement, — • — strain);
(d) Strain PSD.
Next, for the thermally buckled plate
Figure 4(a): The primary strain fr=227 Hz should be compared with the
theoretical displacement /^=279 Hz of Eq. (13). A second spectral energy
peak is found at 537 Hz, much larger than twice (-2.37) the primary strain /^.
Figure 4rb): Here, the spectral energy buildup is most significant at zero
frequency. Besides, there appear two spectral energy humps at 100 and 183
Hz, below the primary strain = 227 Hz of Fig. 4(a). Discounting the zero-
frequency spectral peak, PSD may be approximated by a straight line in the
semi-log plot, hence it is of an exponential form up to 400 Hz.
Figure 4rc): The zero-frequency peak is followed by a single spectral energy
hump at 115 Hz. Again, PSD can be approximated by a straight line and its
slope is roughly the same as in Fig. 4(b).
Figure 4rd): A major spectral energy peak emerges at 130 Hz, followed by a
minor one at 350 Hz. Theoverall spectral energy level is raised so that the
magnitude of PSD ranges over only two decades in the figure.
In Figs. 4(b)-(d) we have ignored the spectral energy peaks at around 500
Hz, for they are not related to the first plate mode under consideration. This is
further supported by the simulation evidence to be discussed presently.
4.2 Numerical simulation results
After choosing .y = 0 or 1.7, we
are left with SPL yet to be specified. o
Ideally, one would like to carry out the
numerical simulation of Eq. (4) by
using SPL of the plate experiment 'm-z
(Table III) and thus generate strain ^
PSDs which are in agreement with
Figs. 3 and 4. Not surprisingly, the _4
reality is less than ideal. An obvious
reason that this cannot be done is that
the forcing energy input is fed into all f
plate modes being excited in Hg. 6 PSD averaged over 12 frequency
experiment, whereas the forcing (j=0, SPL=138dB)
energy excites only one mode in the - displacement; -•-strain
numerical simulation. Consequently, SPL for the numerical simulation should
be less than the experimental SPL, but we do not know a priori how much
less. We therefore choose a SPL to bring about qualitative agreements
between the single-mode simulation and multimode experiment. As anticipated,
the simulation SPLs (Table HI) are consistently smaller than the experimental
values.
The numerical simulation results are shown in Figs. 5-6 for 5 = 0 and Figs.
7-10 for s =1.7. Actually each figure has four frames, denoted by (a)-(d).
First, frames (a) and (b) depict and Since they are very
jagged at large SPLs, we average the spectral energy over 12 frequency
intervals and present both of the smoothed-out frequency response functions in
the same frame (c). Lastly, frame (d) shows Og(/) itself Since there is no
qualitative difference between <E>g(/) and we shall call them both the
strain PSD. We present all four frames (a)— (d) of Figs. 5 and 7, but only the
frame (c) of Figs. 6, 8, 9 and 10 here for the lack of space.
First, for the nonthermal plate
Figure 5: The simulated is closely approximated by Eq. (11) with f =
236 Hz. Note that is also approximated by Eq. (11) for all frequencies
913
Fig. 7 Numerical simulation results under j=1.7 and SPL=129 dB.
(a) Displacement ( - simulation,* Eq. (13)); (b) Strain ( - simulation, • Eq. (13));
(c) PSD averaged over 12 frequency intervals ( - displacement, — • strain);
(d) Strain PSD.
but zero and 476 Hz, where the strain spectral energy piles up due to the
quadratic transformation (9). Since 476 Hz is nearly twice (-2.02) the primary
/^, strain spectral energy buildups are due to the sum and difference of two
nearly equal frequencies, ± /2, where/i==/2^/^.
Figure 6: The primary strain is shifted slightly upward to 253 Hz and the
spectral width at half resonance peak is 50% wider than that of Fig. 5(c). The
spectral energy builds up at 525 Hz which is roughly twice (-2.08) the /^. At
SPL=138 dB we find that the strain spectral energy hump at 525 Hz is about 2
decades below the resonance frequency peak, as was in Fig. 3(b).
Now, for the thermally buckled plate
Figure 7: The simulated and are weU approximated by Eq.
(13) around /^=270 Hz which is a litde below the linearized /^=279 Hz.
Unlike in Fig. 5 for 5=0, both and l/7^(/)F show spectral energy
building up significantly near zero and 543 Hz which is twice (-2.01) the /^.
914
2p — ! - 1 - ! - 1 - 1 - T— n - 1 - r
^ _ 1 - 1 - 1 - < - 1 - > - 1 - ^ - 1
0 300 600
f
Fig. 8 PSD averaged over 12 frequency
intervals (j =1.7, SPL=138 dB)
- displacement; — • — strain
Note that in Fig. 7(a) the spectral
energy hump at 543 Hz is about 3
decades below the primary frequency
peak, as was in Fig. 4(a).
Figure 8: After a large zero-frequency
peak, two spectral energy humps
appear at 131 Hz and 236 Hz. Note
that the ratios of these frequencies to
the /, (131/279 -0.47 and 236/279 -
0.85) are comparable with the same
ratios (100/227 -0.44 and 183/227 -
0.81) found in Fig. 4(b). Excluding
the zero-frequency peak, the overall
strain PSD is a straight line, hence of
an exponential form, as in Fig. 4(b),
Figure 9: The zero-frequency spectral peak is followed by a single major
energy hump at 154 Hz. The ratio of this to the (154/279 -0.56) is
somewhat larger than the ratio (115/227 -0.51) in Fig. 4(c). The strain PSD
can also be approximated by a straight line over the entire frequency range, and
Figs. 8 and 9 seem to have the same slope when fitted by straight lines.
Figure 10: The spectral magnitude of is larger than that of in
the frequency range above 300 Hz. The choice of SPL=146 dB was based on
that the PSD magnitude around 300 Hz is about 2 decades below the main
spectral peak magnitude at 180 Hz, thus emulating Fig. 4(d).
All in all, by numerical simulations we have successfully reproduced the
peculiar features in the two sets of strain PSDs observed experimentally under 5
= 0 and 1.7.
Fig. 9 PSD averaged over 12 frequency
intervals (s =1.7, SPL=143 dB)
- displacement; — • — strain
Fig. 10 PSD averaged over 12 frequency
intervals (j =1.7, SPL=146 dB)
- displacement; — • — strain
915
5. Concluding remarks
At low SPL the nonthermal {s= 0) and post-buckled (^=1.7) plates appear
to have a similar PSD. However, this appearance is quite deceptive in that the
nonthermal plate motion is in a single-well potential, so that PSD does not
change qualitatively as SPL is raised. On the other hand, the trajectory of a
post-buckled plate is in one of the two potential energy wells when SPL is very
small. However, as we raise SPL such a plate motion can no longer be
contained in a potential well, and hence it encircles either one or both of the
potential wells in an erratic manner. This is why the experimentally observed
and numerically simulated strain PSDs of a post-buckled plate exhibit
qualitative changes with the increasing SPL, and thereby reflect the erratic
snap-through plate motion. A quantitative analysis of snap-through dynamics
will be presented elsewhere.
Lastly, we wish to point out that a PSD of straigh-line form in the semi-log
plot was observed in a Holmes oscillator when trajectories are superposed
randomly near the figure-eight separatrix [23].
Acknowledgments
Correspondence and conversations with Chung Fi Ng, Chuh Mei, Rimas
Vaicaitis, and Jay Robinson are sincerely appreciated. We also wish to thank
the referees for their helpful suggestions to improve the readability of this
paper.
References
1. Lee, J., Large-Amplitude Plate Vibration in an Elevated Thermal
Environment, WL-TR-92-3049, Wright Lab., Wright-Patterson AFB, OH,
June, 1992.
2. Jacobson, M.J. and Maurer, O.F., Oil canning of metallic panels in
thermal-acoustic environment, AIAA Paper 74-982, Aug., 1974.
3. Jacobson, M.J., Sonic fatigue of advanced composite panels in thermal
environments, J. Aircraft, 1983, 20, 282-288.
4. Bisplinghoff, R.L. and Pian, T.H.H., On the vibrations of thermally
buckled bars and plates, in Proc. 9th Inter. Congr. of Appl. Mech.,
Brussels, 1957, 7, 307-318.
5. Tseng, W.-Y., Nonlinear vibration of straight and buckled beams under
harmonic excitation, AFOSR 69-2157TR, Air Force Office of Scientific
Research, Arlington, VA, Nov., 1969.
6. Seide, P. and Adami, C., Dynamic stability of beams in a combined
thermal-acoustic environment, AFWAL-TR-83-3072, Flight Dynamics
Lab., Wright-Patterson AFB, OH, Oct., 1983.
7. Ng, C.F., Nonlinear and snap-through responses of curved panels to
intense acoustic excitation, /. Aircraft, 1989, 26, 281-288.
8. Robinson, J.H. and Brown, S.A., Chaotic structural acoustic response of a
milled aluminum panel, 36th Structures, Structural Dynamics, and
916
Material Conference, AIAA-95-1301-CP, New Orleans, LA, 1240-1250,
Apr. 10-13, 1995.
9. Ng, C.F. and Clevenson, S. A., High-intensity acoustic tests of a thermally
stressed plate, J, Aircraft, 1991, 28, 275-281..
10. Ng, C.F. and Wentz, K.R., The prediction and measurement of thermo¬
acoustic response of plate structures, 31st Structures, Structural
Dynamics, and Material Conference, AIAA-90-0988-CP, Long Beach,
CA, 1832-1838, Apr. 2-4, 1990.
11. Holmes, P., A nonlinear oscillator with a strange attractor, Phil. Trans.,
Roy. Soc. of London, 1979, 292A, 419-448.
12. Dowell, E.H. and Pezeski, C., On the understanding of chaos in Duffings
equation including a comparison with experiment, J. Appl. Mech., 1986,
53, 5-9.
13. Lee, J., Random vibration of thermally buckled plates: I Zero temperature
gradient across the plate thickness, in Progress in Aeronautics and
Astronautics, 168, Aerospace Thermal Structures and Materials for a
New Era, Ed. E.A. Thornton, AIAA, Washington, DC, 1995. 41-67.
14. Lee, J., Random vibration of thermally buckled plates: n Nonzero
temperature gradient across the plate thickness, to appear in J. Vib. and
Control, 1997.
15. Lee, J., Large-amplitude plate vibration in an elevated thermal
environment, Mech. Rev., 1993, 46, S242-254.
16. Shinozuka, M. and Jan, C.-M., Digital simulation of random processes
and its applications, J. Sound and Vib. 1912, 25, 1 1 1-128.
17. Chiu, H.M. and Hsu, C.S., A cell mapping method for nonlinear
deterministic and stochastic systems - Part II: Examples of application, J.
Appl. Mech., 1986, 53, 702-710.
18. Vaicaitis, R., Nonlinear response and sonic fatigue of national aerospace
space plane surface panels, J. Aircraft, 1994, 31, 10-18.
19. Vaicaitis, R., Response of Composite Panels Under Severe Thermo-
Acoustic Loads, Report TR-94-05, Aerospace Structures Information and
Analysis Center, Wright-Patterson AFB, OH, Feb., 1994.
20. Kasdin, N.J., Runge-Kutta algorithms for the numerical integration of
stochastic differential equations, J. Guidance, Control, and Dynamics,
1995,18, 114-120.
21. Shampine, L.F. and Gordon, M.K., Computer solution of ordinary
differential equations, 1975, Freeman, San Francisco.
22. Lin, Y.K., Probabilistic theory of structural dynamics, Robert E. Krieger
Publishing, 1976, Huntington, NY.
23. Brunsden, V., Cortell, J. and Holmes, P.J., Power spectra of chaotic
vibrations of a buckled beam, J. Sound and Vib., 1989, 130, 1-25.
917
918
ENHANCED CAPABELITIES OF THE NASA LANGLEY
THERMAL ACOUSTIC FATIGUE APPARATUS
Stephen A. Rizzi and Travis L. Turner
Structural Acoustics Branch
NASA Langley Research Center
Hampton, VA 23681-0001
ABSTRACT
This paper presents newly enhanced acoustic capabilities of the Thermal
Acoustic Fatigue Apparatus at the NASA Langley Research Center. The
facility is a progressive wave tube used for sonic fatigue testing of aerospace
structures. Acoustic measurements for each of the six facility configurations
are shown and comparisons with projected performance are made.
INTRODUCTION
The design of supersonic and hypersonic vehicle stmctures presents a
significant challenge to the airframe analyst because of the wide variety and
severity of environmental conditions. One of the more demanding of these is
the high intensity noise produced by the propulsion system and turbulent
boundary layer [1]. Complicating effects include aero-thermal loads due to
boundary layer and local shock interactions, static mechanical preloads, and
panel flutter. Because of the difficulty in accurately predicting the dynamic
response and fatigue of structures subject to these conditions, experimental
testing is often the only means of design validation. One of the more common
means of simulating the thermal- vibro-acoustic environment is through the use
of a progressive wave tube. The progressive wave tube facility at NASA
Langley Research Center, known as the Thermal Acoustic Fatigue Apparatus
(TAFA), has been used in the past to support development of the thermal
protection system for the Space Shuttle and National Aerospace Plane [2]. It
is presently being used for sonic fatigue studies of the wing strake
subcomponents on the High Speed Civil Transport [3].
The capabilities of the TAFA were previously documented by Clevenson and
Daniels [4]. The system was driven by two Wyle WAS 3000 airstream
modulators which provided an overall sound pressure level range of between
125 and 165 dB and a useful frequency range of 50-200 Hz. A 360 kW quartz
lamp bank provided radiant heat with a peak heat flux of 54 W/cm^. A
schematic of the facility is shown in Figure 1. Representative spectra and
coherence plots are shown in Figures 2 and 3. Since that time, the facility has
undergone significant enhancements designed to improve its acoustic
capabilities; the heating capabilities were not changed. The objectives of the
enhancements were to increase the maximum overall sound pressure level
(OASPL) to 178 dB, increase the frequency bandwidth to 500 Hz and improve
the uniformity of the sound pressure field in the test section. This paper
919
documents the new capabilities of the TAFA and makes comparisons with the
projected performance.
Figure 1: Schematic of the old TAFA facility.
Figure 2: Test section spectra of the Figure 3: Test section coherence of the
old TAFA facility. old TAFA facility.
FACILITY DESCRIPTION
In order to meet the design objectives, extensive modifications were made to
the sound generation system and to the wave tube itself. A theoretical increase
of 6 dB OASPL was projected by designing the system to utilize eight WAS
3000 air modulators compared to the two used in the previous system. A
further increase of nearly 5 dB was expected by designing the test section to
accommodate removable water-cooled insert channels which reduced its cross-
sectional area from 1.9m x 0.33m to 0.66m x 0.33m. The frequency range was
increased through the use of a longer horn design with a lower (15 Hz vs. 27
Hz in the old facility) cut-off frequency, use of insert channels in the test
section to shift the frequency of significant standing waves above 500 Hz, and
design of facility sidewall stmctures with resonances above 1000 Hz. The
uniformity of the sound pressure field in the test section was improved through
several means. A new, smooth exponential horn was designed to avoid the
impedance mismatches of the old design. To minimize the effect of
uncorrelated, broadband noise (which develops as a byproduct of the sound
920
generation system), a unique design was adopted which allows for the use of
either two-, four-, or eight-modulators. When testing at the lower excitation
levels for example, a two-modulator configuration might be used to achieve a
lower background level over that of the four- or eight-modulator
configurations. In doing so, the dynamic range is extended. Lastly, a catenoidal
design for the termination section was used to smoothly expand from the test
section.
Schematics of the facility in the three full test section configurations are shown
in Figures 4-6. In the two-modulator configuration, the 2 x 4 transition cart
acts to block all but two of the eight modulators. The facility is converted
from the two- to four-modulator configuration by the removal of the 2 x 4
transition cart and connection of two additional modulators. In doing so, the
modulator transition cart slides forward and thereby maintains the continuous
exponential expansion of the duct. In the four-modulator configuration, the 4
X 8 transition cart acts to block the two upper and two lower modulators.
Removal of this component and connection of the four additional modulators
converts the facility to the eight-modulator configuration. Again, the
continuous exponential expansion is maintained as the modulator transition
cart slides forward.
Figure 4: Two-modulator full test section configuration.
921
Figure 5: Four-modulator full test section configuration.
Figure 6: Eight-modulator full test section configuration.
Schematics of the three reduced test section configurations are shown in
Figures 7-9. In these configurations, the horn cart is discarded and the horn
transition cart mates directly to the test section. Water-cooled inserts are used
in the test section to reduce its cross-sectional area. Upper and lower inserts in
the termination section are used to smoothly transition the duct area to the full
dimension at the exit. Conversion from the two- to the four-modular
configuration and from the four- to the eight-modulator configuration is again
accomplished through removal of the 2 x 4 and 4x8 transition carts,
respectively.
922
HORN TTUNSTTION SECTION
Figure 7: Two-modulator reduced test section configuration.
Figure 8: Four-modulator reduced test section configuration.
TEST PROCEDURE
Measurements were taken for several conditions in each of the six facility
configurations. Each modulator was supplied with air at a pressure of 207 kPa
(mass flow rate of approximately 8.4 kg/s) and was electrically driven with the
same broadband (40-500 Hz) signal. Acoustic pressures were measured at
several locations along the length of the progressive wave tube using B&K
model 4136 microphones and Kulite model MIC-190-HT pressure transducers,
see Table 1. The positive x-direction is defined in the two-modulator full
configuration (from the modulator exit) along the direction of the duct. The
positive y-direction is taken vertically from the horizontal centerline of the
923
HORN TRIWSmOH SECTION
MODULATOR TRANSITION
FLEMSie HOSE
adapter puts assembly
TEST SECTION
ADAPTER plate assembly
TERM1ASAT10N SECTION
horn TRANSITION CART
Figure 9: Eight-modulator reduced test section configuration.
duct and the positive z-direction is defined from the left sidewall of the duct as
one looks downstream.
Table 1: Kulite (K) and microphone (M) locations of acoustic measurements.
Loc.
Description
Type
Coordinate (m)
1
Test Sect. Horizontal Centerline Upstream
K
7.75, 0, 0
2
Test Sect. Horizontal Centerline Downstream
K
8.71, 0, 0
5
Test Sect. Vertical Centerline Top
M
8,23, 0.3, 0
15
Test Sect. HorizontaWertical Centerline
K
8,23, 0, 0
25
Test Sect. Vertical Centerline Bottom
M
8.23, -0.3, 0
28
2x4 HorizontaWertical Centerline
M
2.19, 0,0
29
4x8 Horizontal Centerline, % Downstream
M
3.66, 0, 0
30
Horn Tran, Hor. Centerline, % Downstream
M
4.75, 0, 0
35
Termination HorizontaWertical Centerline
M
12.46, 0, 0.17
The acoustic pressure at location 1 was used as a reference measurement for
shaping the input spectrum and for establishing the nominal overall sound
pressure level for each test condition. For each configuration, the input
spectrum to the air modulators was manually shaped through frequency
equalization to produce a nearly flat spectrum at the reference pressure
transducer. Data was acquired at the noise floor level (flow noise only) and at
overall levels above the noise floor in 6 dB increments (as measured at the
reference location) up to the maximum achievable. Thirty-two seconds of
time data were collected at a sampling rate of 4096 samples/s for each
transducer in each test condition. Post-processing of the time data was
performed to generate averaged spectra and coherence functions with a 1-Hz
frequency resolution.
924
RESULTS
For each facility configuration, plots of the following quantities are presented:
normalized input spectrum to the air modulators, minimum to maximum
sound pressure levels at the reference location, maximum sound pressure
levels in the test section, maximum sound pressure levels upstream and
downstream of the test section, and vertical and horizontal coherence in the
test section. The minimum levels in each case correspond to the background
noise produced by the airflow through the modulators.
Normalized input voltage spectra to each modulator for each configuration are
shown in Figures 10, 15, 20, 25 and 30. These spectra were generated to
achieve as flat an output spectrum as possible at the reference location for the
frequency range of interest (40-200 Hz for the full section, 40-500 Hz for the
reduced section). As expected, the significant difference between the full and
reduced configurations is seen in the high (>200 Hz) frequency content.
Figure 11 shows a background noise level of 126 dB (the lowest of all
configurations) for the two-modulator full test section configuration. Nearly
flat spectra are observed below 210 Hz for levels above 130 dB, giving a
dynamic range of about 32 dB. The flat spectrum shape is a significant
improvement over the performance of the old configuration as shown in Figure
2. Standing waves are evident at frequencies of 210, 340 and 480 Hz. For this
reason, the full section operation is limited to less than 210 Hz or to the 220-
330 and 370-480 Hz frequency bands. The effect of standing waves are
explored in further depth in the next section. The spectra in Figure 12 indicate
a nearly uniform distribution in the x-direction throughout the test section. It is
interesting to note that Figure 13 shows no sign of standing waves upstream of
the test section, confirming that the cause is associated with the test section.
Lastly, a near perfect coherence between upstream and downstream, and upper
and lower test section locations is shown in Figure 14 for frequencies between
40 and 210 Hz. Again, this is a significant improvement over the performance
of the old configuration (Figure 3).
Figure 10: Normalized input spectrum Figure 11: Min to max SPL at location
(2-modulator full). 1 (2-modulator full).
925
Figure 12: SPL in test section at max Figure 15: Normalized input spectrum
level (2-modulator full). (4-modulator full).
Figure 13: SPL along length of TAFA Figure 16: Min to max SPL at location
(2-modulator full). 1 (4-modulator full).
Figure 14: Test section coherence (2- Figure 17: SPL in test section at max
modulator full). level (4-modulator full).
The four-modulator full configuration exhibits similar behavior as the two-
modulator full configuration as seen in Figures 16-19. The lowest level at
which a uniform spectrum is achieved is 137 dB, giving a dynamic range of
roughly 30 dB in this configuration. Lastly, the eight-modulator full
926
configuration results, shown in Figures 21-24, indicate a noise floor of about
142 dB and dynamic range of 22 dB.
Frequency, Hz Frequency, Hz
Figure 18: SPL along length of TAFA Figure 21: Min to max SPL at location
(4-modulator full). 1 (8-modulator full).
Frequency, Hz Frequency, Hz
Figure 19: Test section coherence (4- Figure 22: SPL in test section at max
modulator full). level (8-moduiator full).
Frequency, Hz Frequency, Hz
Figure 20: Normalized input spectrum Figure 23: SPL along length of TAFA
(8-modulator full). (8-modulator full).
927
Frequency, Hz Frequency, Hz
Figure 24: Test section coherence (8- Figure 27: SPL in test section at max
modulator full). level (2-modulator reduced).
Figure 25: Normalized input spectrum Figure 28: SPL along length of TAFA
(2-modulator reduced). (2-modulator reduced).
Figure 26: Min to max SPL at location Figure 29: Test section coherence (2-
1 (2-modulator reduced). modulator reduced).
The reduced test section configurations are used to increase the frequency
range and maximum sound pressure level in the test section. Results for the
two-modulator reduced configuration, shown in Figures 26-29, indicate a
nearly flat spectrum between 40 and 480 Hz, a noise floor of 129 dB and a
dynamic range of about 28 dB. Coherence in the test section is nearly unity
928
over this frequency range. This represents a significant improvement over the
old facility configuration. Results of similar quality indicate a d5mamic range
of roughly 26 and 29 dB for the four- (Figures 31-34) and eight-modulator
(Figures 36-39) configurations, respectively. Note that the coherence for these
configurations is slightly reduced at the high frequencies, but is still very good
out to 480 Hz.
Figure 30: Normalized input spectrum Figure 33: SPL along length of TAFA
(4-modulator reduced). (4-modulator reduced).
160 r
80
OASPLs: 134.1, 135.5, 142.1, 147.6,
153.9, 160.1, 165.9, 167.9
1.0
! 1
1 1
Loc 1, Loc 2
Loc 5, Loc 25
400
100
200 300
Frequency, Hz
400
500
Figure 3 1 : Min to max SPL at location Figure 34: Test section coherence (4-
1 (4-modulator reduced). modulator reduced).
Figure 32: SPL in test section at max Figure 35: Normalized input spectrum
level (4-modulator reduced). (8-modulator reduced).
929
80
OASPLs; 134.1, 135.7, 141.9, 148.3,
154.1. 160.0, 165.9. 170.5
80
Loc30
Loci
100 200 300 400 500 100 200 300 400 500
Frequency. Hz Frequency, Hz
Figure 36: Min to max SPL at location Figure 38: SPL along length of TAFA
i (8-modulator reduced). (8-modulator reduced).
Figure 37: SPL in test section at max Figure 39: Test section coherence (8-
level (8-modulator reduced). modulator reduced).
Table 2 presents a summary of the maximum average OASPL for each facility
configuration. In each case, the number of active modulators were run at
maximum power as an independent group (independently for the single
modulator case) and the results averaged. For example, results for one active
modulator were obtained by running each modulator individually and
averaging the resulting pressures.
Table 2: Summary of maximum average overall sound pressure levels (dB).
Number of Active Modulators
1
2
4
8
2-Modulator Red.
1
1 j
2-Modulator Full
i
4-Modulator Red.
159.1
—
MSM
4-Modulator Full
155.6
161.2
8-Modulator Red.
158.4
—
mmm
171.7*
8-Modulator Full
153.0
158.4
164.5
170.0
^Pressure scaled by ^7? from 7-modu
ator run
DISCUSSION
In this section, limiting behaviors of the full and reduced test section
configurations are explored and the effect of test section inserts, modulator
coupling and wave tube performance are discussed.
Limiting Behaviors
The auto-spectra from the full test section configurations exhibit sharp
reductions in level at approximately 210, 340, and 480 Hz. This behavior
corresponds to measurements near nodes of vertical (height) standing waves in
the test section portion of the wave tube. Table 3 summarizes theoretical,
resonant frequencies and corresponding modal indices of the test section duct
resonances within the excitation bandwidth. The modal indices m and n
correspond to half wavelengths in the vertical and transverse (width) directions
of the cross section, respectively. There are several resonances that may be
excited below 500 Hz, but only three of these appear to be significant at the
test section transducer locations (about the horizontal centerline). Because of
the presence of air flow in the facility and lack of measurements in the cross
section, it is difficult to correlate the experimental and theoretical modes.
Measurements of the acoustic pressure at several locations in a cross-section of
the duct will be necessary to fully characterize the resonant behavior. It is
sufficient to say that the usable frequency range in the full test section
configurations is approximately 40-210 Hz near the horizontal centerline.
Acoustic pressure auto-spectra from the reduced test section configurations are
essentially flat to almost 500 Hz. This is due to the fact that only two
resonances are within the excitation bandwidth for this configuration, see
Table 3. A sharp reduction is noted in the vicinity of 480 Hz. Although the
(m=l, n=0) resonance does not appear to be significant, close inspection of the
data (not shown) indicates its presence. Therefore, the usable frequency range
for the reduced test section configurations is approximately 40-500 Hz.
Table 3: Theoretical resonant frequencies of test section duct modes in Hz.
Performance of Test Section Configurations
For constant input acoustic power, the change from full to reduced test section
configurations should theoretically result in a 4,7 dB increase in OASPL.
However, Table 2 shows that increases of only 2.1 (e.g. 164.3-162.2), 0.9, and
1.7 dB were realized for the two-, four- and eight-modulator configurations.
The system efficiency (actual/expected mean-square pressure) of the two-,
four- and eight-modulator reduced configurations is 38, 40 and 44 percent,
respectively, compared with 51, 63 and 62 percent for the two-, four- and
eight-modulator full configurations. The expected pressure is calculated based
upon a input-scaled value of the rated acoustic power of the WAS 3000
modulator assuming incoherent sources (3 dB per doubling). In general, the
full section efficiency is greater than the corresponding reduced section
efficiency. While the reason for this phenomena is not known, it is
conjectured that the lack of expansion in the reduced configurations limits the
development of plane waves. Therefore, phase and amplitude mismatches
between acoustic sources may be accentuated.
Modulator Coupling Performance
A simplified waveguide analysis for coherent, phase-matched sources predicts
increases in OASPL as shown in Table 4. Measured performance gains were
less than predicted because of the assumptions of the waveguide analysis
(inactive source area treated as hard wall), and possible reductions due to
phase differences between modulators and non-parallel wave fironts at the exit
of the modulator cart, see Figures 4-9. The latter effect is due to different
angles of inclination of the sources relative to the axis of the wave tube. The
greater gains achieved in the full test section configurations support the above
contention that they are more efficient than the reduced configurations in
combining the acoustic sources.
Table 4: Change in SPL (dB) from 1 to max. number of active modulators.
Configuration
A SPL fi:om 1 Active Mod. (Meas/Pred)
2-Modulator Red. (2 active mods.)
3.9/6.53
2-Modulator Full (2 active mods.)
5.5/6.53
4-Modulator Red. (4 active mods.)
8.8 / 13.98
4-Modulator Full (4 active mods.)
11.4/13.98
8-Moduiator Red. (8 active mods.)
13.3/22.10
8-Moduiator Full (8 active mods.)
17.0/22.10
Wave Tube Performance
A change in configuration from the two- to the four-modulator configurations,
and from the four- to the eight-modulator configurations, will result in an
incremental increase of 3 dB in OASPL if the individual sources are phase-
matched. This is due to a pure doubling of the power without any change in
the radiation impedance of the individual sources. For the reduced
configurations, a 3.6 and 3.8 dB increase are observed, respectively. A 4.8 and
932
3.0 dB increase are observed for the full configurations, respectively. Note
that a greater than 3 dB increase is possible when the higher modulator
configuration (for example, the four-modulator reduced configuration) is less
susceptible than the lower modulator configuration (the two-modulator
reduced configuration) to phase mismatches between modulators. This seems
plausible because any such mismatches are averaged over a larger number of
sources.
SUMMARY
Modifications to the NASA Langley TAFA facility resulted in significant
improvements in the quality and magnitude of the acoustic excitation over the
previous facility. The maximum OASPL was increased by over 6 dB (vs the
previous 165 dB) with a nearly flat spectrum between 40-210 and 40-480 Hz
for the full and reduced test section configurations, respectively. In addition,
the coherence over the test section was excellent. These improvements,
however, did not meet the objective for a maximum OASPL of 178 dB.
There are several reasons why the maximum OASPL did not meet the
objectives, including a lack of expansion in the reduced configurations and
phase differences between modulators. A detailed computational analysis
would be desirable to indicate the source of the inefficiencies and to help
identify possible means of increasing the overall system performance.
ACKNOWLEDGEMENTS
The authors wish to thank Mr. H. Stanley Hogge and Mr. George A. Parker for
their support in configuring and running the facility. We wish to also thank
Mr. James D. Johnston, Jr. of NASA Johnson Space Center for loan of four
Wyle air modulators.
REFERENCES
1. Maestrello, L., Radiation from a Panel Response to a Supersonic
Turbulent Boundary Layer, Journal of Sound and Vibration, 1969,
10(2), pp. 261-295.
2. Pozefsky, P., Blevins, R.D., and Langanelli, A.L., Thermal-Vibro-
Acoustic Loads and Fatigue of Hypersonic Flight Vehicle Structure,
AFWAL-TR-89-3014,
3. Williams, L.J., HSCT Research Gathers Speed, Aerospace America,
April 1995, pp. 32-37.
4. Clevenson, S.A. and Daniels, E.F., Capabilities of the Thermal
Acoustic Fatigue Apparatus, NASA TM 104106, February 1992.
933
SONIC FATIGUE CHARACTERISTICS OF HIGH TEMPERATURE MATERIALS AND
STRUaURES FOR HYPERSONIC FLIGHT VEHICLE APPLICATIONS-
Dr. I. Holehouse, Staff Specialist,
Rohr Inc., Chula Vista, California
1. INTRODUCTION SUWiARY
A combined analytical and experimental program was conducted to investigate
thermal -acoustic loads, structural response, and fatigue characteristics of
skin panels for a generic hypersonic flight vehicle. Aerothermal and
aeroacoustic loads were analytically quantified by extrapolating existing
data to high Mach number vehicle ascent trajectories. Finite-element
thermal and sonic fatigue analyses were performed on critically affected
skin panels. High temperature random fatigue shaker tests were performed
on candidate material coupons and skin-stiffener joint subelements to
determine their random-fatigue strength at high temperatures. These were
followed by high temperature sonic fatigue tests of stiffened-skin panels
in a progressive wave tube. The primary materials investigated were
carbon-carbon and silicon-carbide refractory composites, titanium metal
matrix composites and advanced titanium alloys. This paper reports on the
experimental work and compares measured frequencies and acoustically
induced response levels with analytically predicted values.
The coupon shaker test data were used to generate material random fatigue
"S-N" curves at temperatures up to 980°C. The joint subelements provided
data to determine the effects on fatigue life of skin-stiffener joining
methods. The PWT sonic fatigue panel tests generated response and fatigue
life data on representative built-up skin panel design configurations at
temperatures up to 925“C and sound pressure levels up to 165 dB. These^
data are used in determining the response strains and frequencies of skin
panel designs when subjected to combined thermal -acoustic loading and to
identify modes of failure and weaknesses in design details that affect _
sonic fatigue life. Sonic fatigue analyses of selected test panel design
configurations using finite-element techniques were also performed and
related to the experimental results. Acoustically induced random stresses
were analytically determined on a mode-by-mode basis using finite element
generated mode shapes and an analytical procedure that extends Miles
approach to include multi-modal effects and the spatial characteristics of
both the structural modes and the impinging sound field.
The paper also describes the instrumentation development work performed in
order to obtain reliable strain measurements at temperatures in excess of
conventional strain gauge capabilities. This work focused primarily on the
use of recently developed high temperature (350"C to 1000“C) strain gauges,
laser Doppler vibrometers, high temperature capacitance displacement
probes, and the determination of strain-displacement relationships to
facilitate the use of double integrated accelerometer data to derive strain
levels.
935
This work was funded by the USAF Flight Dynamics Laboratory (Kenneth R.
Wentz, Project Engineer). The complete program report is contained in
References 1 and 2.
2. HIGH TEMPERATURE STRAIN MEASUREMENTS
Conventional adhesively bonded strain gauge installations are temperature
limited to approximately 350°C. In order to achieve strain measurements at
higher temperatures, up to QSO^C, ceramic layers and coatings were used to
both attach strain gauges and to thermally protect them. However, such
strain gauge installations are very sensitive to process parameters which
often need varying depending upon the test specimen material. Coated
carbon-carbon is a particularly difficult material to adhere to due to its
material characteristics and relatively rough surface texture. Carbon-
carbon also has a near zero coefficient of thermal expansion which presents
attachment and fixturing problems in a high temperature environment.
When high test temperatures either preclude or make problematic the use of
strain gauges, an alternative technique for obtaining strain levels is to
measure displacements and then determine strain levels using strain-
displacement ratios. Strain is directly proportional to displacement for a
given deflected shape, or mode shape, regardless of changes in the elastic
modulus of the specimen material as it is heated. Consequently, if the
deflected shape does not change significantly with temperature, high
temperature test strain levels can be determined from room temperature
strain and displacement measurements in combination with displacement
measurements made at the test temperature.
This measurement technique facilitates the use of non- contacting
transducers which can be located away from the heated area, such as
capacitance displacement probes or Laser Doppler Vibrometers (LDV). LDVs
actually measure surface velocity but their signal outputs can be readily
integrated and displayed as displacement. Accelerometers can also be used
to measure displacement by double integrating their signal output.
However, since accelerometers require surface contact they have to either
withstand, or be protected from, the thermal environment. When this is not
readily achievable, it is sometimes possible to install an accelerometer at
a location on the test specimen or fixturing where the temperature is
within its operating range, providing the displacement response at the
point of measurement is fully coherent with the strain response at the
required location.
The displacement range limitations of the LDV and capacitance probes
available to the program resulted in having to use double-integrated
accelerometer outputs to measure displacements at room temperature and at
the test temperature. Conventional strain gauges were used to measure
strains at room temperature. In order to confirm that the strain-
displacement ratios were unaffected by temperature, limited high
temperature strains were measured at temperatures up to 980* C. Once the
strain-displacement ratio for a given specimen type was determined, air¬
cooled accelerometers were used to determine high temperature test strain
levels. The level of measurement accuracy of this technique was estimated
to be within 10 percent.
936
The most successful strain measurements made at 980“ C utilized a ceramic
flame spray installation of an HFN type free filament gauge. This gauge
installation included the use of silicon-carbide (SiC) cement as a base
coat for the gage, applied over a 1-inch square area of a lightly sanded
carbon-carbon surface substrate. Lead wire attachments to the gauge were
made with standard Nichrome ribbon wire anchored to the specimen with SiC
cement. With this gauge installation, it was possible to make dynamic
strain measurements for short periods of time at 980“C.
3. RANDOM FATIGUE SHAKER TESTS
The instrumented test specimens were mounted in a duckbill fixture and the
specimen/fixture assembly then enclosed in a furnace. An opening in the
furnace allows the specimen tip to protrude out in order to accorrmodate the
air-cooled tip accelerometer. Figure 1 shows strain gauge locations and
fixturing for material coupon and joint subelement specimens.
The test procedure comprised a room temperature sine-sweep in order to
identify the fundamental mode and its natural frequency, one-third octave
random loading at room temperature centered around the fundamental natural
frequency and one-third octave random endurance testing at the required
test temperature and load level.
Twelve inhibited carbon-carbon material coupons generated usable S-N data,
eleven at 980“C and one at 650“C. S-N data points were also generated at
980°C for two integral joint and two mechanically fastened joint
subelements. Fixturing problems and specimen availability limited the
number of S-N data points generated. Figure 2 shows the random fatigue S-N
data points with joint subelement data points superimposed. The random
fatigue endurance level for the material coupons, extrapolated from 10 to
10® cycles, is approximately 320 microstrain rms. The integral joint
subelements did not fail at the strain gauge locations; consequently, the
actual maximum strain levels were higher than those shown on Figure 2.
Taking this into account, it appears that the integral joints have a
fatigue endurance level of greater than one-half of that for the material
coupons. The mechanically fastened joint subelements exhibited fatigue
strength comparable to that of the material coupons. These results
indicate that carbon-carbon joints and attachments methods are not
critically limiting factors in the structural applications of inhibited
carbon-carbon. Figure 3 shows a representative example of the strain
amplitude and peak strain amplitude probability density functions at room
temperature for a material coupon specimen. The "peak" function can be
seen to approximate a Rayleigh distribution, as it should for a Gaussian
random process.
Random fatigue S-N data were also generated for enhanced silicon-carbide
composites (SiC/SiC) including thermally exposed specimens (160 hours at
980“ C), titanium metal matrix composites (TMC) utilizing Ti 15-3 and Beta
21S titanium matrix materials, titanium aluminide (super alpha two),
titanium 6-2-4-2, titanium 6-2-4-2-$i (including thermally exposed
specimens) and Ti-1100. The fatigue endurance levels are shown in Table 1.
Also shown in Table 1 are S-N data points for uninhibited carbon-carbon
generated on a previous program (Reference 3).
937
TABLE 1. SUMMARY OF RANDOM FATIGUE ENDURANCE LEVELS.
material
TEMPERATURE
ENDURANCE LEVEL CORRESPONDING TO 10®
CYCLES: OVERALL RMS STRAIN (MICROSTRAIN)
material COUPONS
SUBELEMENTS
INHIBITED CARBON-CARBON
1800“F (980"C)
320
integral JOINTS > 160
BOLTED JOINTS 320
5 PLY
1800T (980"C)
100
-
UNINHlbl 1 tU
CARBON-CARBON
1ft PI Y
ISOO'-F igSOT)
150
_
*5 PLY
1000‘'F I540"C)
100
-
18 PLY
lOOO'F (540-C)
450
-
5 PLY
ROOM TEMPERATURE
550
-
18 PLY
ROOM TEMPERATURE
450
-
ENHANCED
SiC/SiC
NON-EXPOSED
1800"F r980“C)
450
-
THERMALLY EXPOSED
iaOO“F f980‘'C)
300
-
1000“F (540*0
520
DIFFUSION-BONDED HAT-
STIFFENED = 520
n IML
ROOM TEMPERATURE
2250
-
beta 21S TMC
200
510
-
TITANIUM ALUMINIDE
/ciinCD ftl DUIA TUn^
ROOM TEMPERATURE
410
{ rvui 1 iri • *
Ti-6242-Si
735
Hilllll
LIO BONDED HONEYCOMB
BEAM = 388
■■
WELDED JOINT = 400
T-; nnn
-
Ti 6-2-4-2
ROOM TEMPERATURE
675
-
866MISC/039-T1.IH
12-02-96
938
Figure 4 shows random fatigue S-N curves for the materials tested
superimposed on one graph for comparison purposes. The Ti 15-3 TMC data
are not shown since this was a concept demonstrator material utilizing a
Ti 15-3 matrix material for producibility reasons. Ti 15-3 does not have
the temperature capability for hypersonic vehicle applications. Titanium
aluminide data are not shown due to its brittle material characteristics
making it unsuitable for sonic fatigue design critical structures. Ti-1100
S-N data were very similar to the non-exposed Ti 6-2-4-2-Si and are not
shown. Ti 6-2-4-2 coupons were only tested at room temperature before
beingVeplaced by Ti 6-2-4-2-Si, which has higher structural temperature
capabilities.
The fatigue curves in Figure 4 show inhibited carbon-carbon to have higher
fatigue strength at 980°C than does its uninhibited counterpart. Inhibited
carbon-carbon also has greater resistance to oxidation at high
temperatures.
Although unexposed enhanced SiC/SiC had greater random fatigue strength at
980“C than did inhibited carbon-carbon, the two materials exhibited similar
strength at temperature after allowing for thermal exposure. However,
SiC/SiC has a maximum temperature capability of 1100 to 1200°C compared to
1700 to 1900“C for carbon-carbon.
The Beta 21S TMC material demonstrated resonable fatigue strength at 815°C
and the Ti 6-2-4-2-Si specimens exhibited high fatigue strength at 620°C to
650“C.
4. SONIC FATIGUE PANEL TESTS
These tests were performed in Rohr's high temperature progressive-wave tube
(PWT) test facility. The facility is capable of generating overal 1^ sound
pressure levels of 165 to 168 dB at temperatures up to 925 “C to 980“ C,
depending upon the test panel configuration and material.
Three rib-stiffened carbon-carbon panels and a monolithic hat-stiffened
Beta 21S TMC panel were subjected to sonic fatigue testing. Response
strains were measured on the four panels over a range of incrementally
increasing sound pressure levels (140 to 165 dB) at room temperature. One
carbon-carbon panel was subjected to sonic fatigue testing at room
temperature and the other two tested at 925“ C. The TMC panel was endurance
tested at 815“C. Figures 5 and 6 show a carbon-carbon panel and its
fixturing installed in the PWT. The panels were attached to the fixture
via flexures in order to allow for differences in the thermal expansion of
the panel and fixture materials. Structural details of the panels and
instrumentation locations are given in References 1 and 2.
The three carbon-carbon panel configurations encompassed two skin
thicknesses and two stiffener spacings as follows:
Panel 1: 3 skin bays, 6 in. by 20 in. by 0.11 in. thick
Panel 2: 2 skin bays, 9 in. by 20 in. by 0.11 in. thick
Panel 3: 3 skin bays, 6 in. by 20 in. by 0.17 in. thick
939
Table 2 summarizes the measured room temperature frequencies and strain
response levels:
TABLE 2. ROOM TEMPERATURE RESPONSE OF TEST PANELS-
TEST
PANEL
FREQUENCY OF
IN-PHASE
MODE (Hz)
OASPL
(dB)
OVERALL RMS STRAINS
rMICROSTRAIN)
EDGE OF
SKIN BAY
CENTER OF
SKIN BAY
CARBON-CARBON NO. 1
267
165
305
149
155 & 171
145
126
39
191
59
165
558*
173*
CARBON-CARBON NO. 3
423
165
69
127
BETA 21$ TMC
241
165
HIGHEST STRAIN = 287
AT PANEL CENTER ON
STIFFENER CAP
* EXTRAPOLATED ON THE BASIS OF TUE STRAIN RESPONSE WITH SPL FOR
PANELS 1 AND 3.
Panel 1 was subjected to 165 dB at room temperature for 10 hours at which
point cracks developed at the ends of the stiffeners. The frequency
dropped slightly during the ten hour test resulting in the number of cycles
to failure being approximately 9 million.
Panel 2 was endurance tested at 925“C at 150, 155 and 160 dB for 3-1/2
hours at each level, followed by one hour at 165 dB. At this point, cracks
were observed at the ends of the stiffeners, similar to the cracks in
Panel 1.
Panel 3 was endurance tested at 925“C and 165 dB for 10 hours without any
damage to the panel.
The TMC panel was endurance tested at 815° C and 165 dB for 3-1/2 hours at
which time cracks were observed in two stiffener caps at the panel center.
The high test temperatures for Panels 2 and 3 and the TMC panel precluded
attaching an accelerometer directly to the panel surface, even with air
cooling. This prevented the direct measurement of panel displacements at
925° C. In order to attempt to estimate the high temperature endurance test
strain levels, a temperature survey was performed on the panel fixturing
with Panel 3 installed in order to determine an acceptable location for an
accelerometer. An accelerometer at the selected fixture location tracked
linearly with the highest reading strain gauges during a room temperature
response survey. The coherence between the fixture accelerometer and the
panel strain gauges was 0.9 in the frequency range of panel response.
940
Having established a coherent strain displacement relationship at room
temperature, the temperature was increased progressively with increasing
acoustic loading, generating accelerometer and microphone data at 480 C and
140 dB, 650°C and 155 dB, 860°C and 155 dB, and 925“C at 165 dB. It was
clear from the data at the higher temperatures and load levels that the
full spectrum overall rms displacement levels obtained by double integrat¬
ing the accelerometer output signals could not be used to determine high
temperature strain levels due to high amplitude, low frequency displace¬
ments (displacement being inversely proportional to frequency squared for a
given "g" level) that were well below the panel response frequency range
and therefore would not be proportional to panel strain levels. It is
important to remember here that since the accelerometer is mounted on the
panel fixture, it is measuring fixture response, some of which is not
related to panel response.
After reviewing the various frequency spectra, it was decided to re-analyze
the data to generate overall rms levels over selected frequency bandwidths
that would encompass a high percentage of the full -spectrum overall rms
strains and eliminate the low frequency displacements. If a consistent
strain-displacement relationship could be established at room temperature
within a frequency bandwidth such that the strains could be related to the
full-spectrum overall rms strains, and if the same bandwidth could be used to
generate displacements at temperatures that were sufficiently consistent to
relate to strain response, then it would be possible to at least make a
reasonable estimate of the test temperature strain level. It was determined
that band-passed response data in the 300 to 600 Hz frequency range gave
consistent strain-displacement ratios at room temperature. Double-integrated
band-passed accelerometer outputs (displacements) were consistent with
increasing sound pressure levels at incrementally increasing test temperatures
up to the 925°C/165 dB endurance test conditions. Table 3 summarizes the high
temperature test panel results.
TABLE 3. HIGH TEMPERATURE TEST PANEL RESULTS.
TEST PANEL
TEST
TEMPERATURE
OVERALL
SOUND
PRESSURE
LEVEL (dB)
HIGHEST
ESTIMATED
OVERALL RMS
STRAIN
(MICROSTRAIN)
EXPOSURE TIME,
ESTIMATED FATIGUE
CYCLES AND COMMENTS
(°F)
rc)
BETA 21S TMC
PANEL
1500
815
165
NOT ESTIMATED
3 1/2 HRS, 3x10“ CYCLES,
STIFFENERS CRACKED AT
MID-SPAN
CARBON-CARBON
PANEL NO. 2
150
155
TTTORSTTTMo^TYOlsr
NO FAILURE
155
219 ^
3 1/2 HRS, 2.3x10° CYCLtS,
NO FAILURE
160
316
3 1/2 HRS. 2.3x10“ CYCLES,
NO FAILURE
165
453
1 HR, 6.4xl0‘> cycles!
CRACKS AT STIFFENER ENDS
925
165
103
10 HRS, 1.7x10' CYCLES,
NO FAILURE
941
It should be noted that carbon-carbon panels 1 and 2 exhibited cracks at
the stiffener ends, whereas the maximum measured strains were at the edges
of skin bays. Consequently, the actual strain levels at the crack
locations were either higher than the measured levels or there were
significant stress concentrations at the stiffener terminations.
5. COMPARISON OF ANALYTICAL AND TEST RESULTS FOR CARBON-CARBON PANELS
MSC NASTRAN was used to perform finite element analyses on the three
carbon-carbon panels that were subjected to the sonic fatigue testing
described in Section 4. The oxidation resistant coating was modeled as a
non-structural mass, which is compatible with the panel test results. ^
Natural frequencies, mode shapes and acoustically induced random strain
levels were analytically determined for room-temperature conditions and
compared to the room-temperature panel test results.
Acoustically induced random stresses were analytically determined on a
mode-by-mode basis using the finite element generated mode shapes and a
Rohr computer code based on an analytical procedure presented in
Reference 4. This procedure extends Miles* approach (Reference 5) to
include multi-modal effects and the spatial characteristics of both the
structural modes and the impinging sound field.
Table 4 shows the calculated and measured frequencies, overall rms strain
levels and the strain spectrum levels for the in-phase stiffener bending
mode for the carbon-carbon panels at room temperature.
TABLE 4. CALCULATED AND MEASURED RESPONSE FREQUENCIES AND STRAIN
LEVELS FOR CARBON-CARBON PANELS AT ROOM TEMPERATURE.
STRAIN LEVELS AT EDGE OF SKIN BAY
NATUF
FREQUE
OF IN-F
MODE
(Hzl
\fKL
:ncy
>HASE
OVERALL RMS STRAIN
(MICROSTRAIN)
STRAIN
SPECTRUM LEVEL
IN-PHASE MODE
(MICROSTRAIN/Hz)
FE ANALYSIS
MEASURED
FE ANALYSIS
MEASURED
FE ANALYSIS
MEASURED
PANEL 1
a65 dBl
305
267
510
305
84
60
PANEL 2
(145 dBl
190
155 &
171
133
126
40
41
PANEL 3
(165 dB)
460
423
77
69
16
16
942
The above results show good agreement between the finite element generated
values and those measured. The level of agreement is particularly good for
the strain spectrum levels, which are typically more difficult to
accurately predict. Figure 7 shows the finite-element frequency solution
for Panel 3. The in-phase mode shape can be seen to have an overall modal
characteristic due to the relatively low bending stiffness of the
stiffeners for the skin thickness used. Figure 8 shows the measured and
finite-element generated strain frequency spectra for Panel 3.
Details of the finite-element analyses and models are contained in
References 1 and 2.
6. CONCLUSIONS AND RECOMMENDATIONS
1. The high temperature testing techniques and strain measuring
procedures successfully generated usable random fatigue S-N
curves and panel response data. The use of strain-displacement
ratios were shown to be an effective alternative to high
temperature strain gauge measurements.
2. In general, the materials and structural concepts tested
demonstrated their suitability for hypersonic flight vehicle skin
panel applications. The major exception was Titanium-Aluminide
Super Alpha Two which was determined to be too brittle.
3. Inhibited carbon-carbon exhibited significantly higher random
fatigue strength at 980°C than did the uninhibited carbon-carbon
— two to three times the random fatigue endurance strain level.
4. Thermally exposed enhanced SiC/SiC had comparable fatigue
strength to that of inhibited carbon-carbon at 980 °C.
5. The TMC specimens usefully demonstrated the fatigue strength of
the TMC concept and the need to develop the concept to
incorporate higher temperature capability titanium matrix
materials.
6. Titanium 6-2-4-2-Si exhibited high fatigue strength in the 590°C
to 650 “C temperature range and also demonstrated the need for TMC
materials to utilize higher temperature matrix materials in order
to be cost effective against the newer titanium alloys.
7. The level of agreement between the finite element analysis
results for the carbon-carbon panels and the progressive-wave
tube test data demonstrated the effectiveness of the analytical
procedure used. The analysis of structures utilizing materials
such as carbon-carbon clearly presents no special difficulties
providing the material properties can be well defined.
943
8 It is recommended that further tests be conducted similar to
those performed in this program but with greater emphasis on
testing panels having dimensional variations in order to develop
design criteria and life prediction techniques. Such testing
should be performed on those structural materials and design
concepts that emerge as the major candidates for flight vehicle
applications as materials development and manufacturing
techniques progress.
REFERENCES
1 R D. Blevins and I. Holehouse, "Thermo-Vibro Acoustic Loads and
rkigue of Hypersonic Flight Vehicle Structure," Rohr, Inc.
Engineering Report RHR 96-008, February 1996.
2. United States Air Force Systems Command, Flight Dynamics
Laboratory Final Technical Report, Contract No. F33615-87-C-33^^/,
to be published.
3. R. D. Blevins, "Fatigue Testing of Carbon-Carbon Acoustic Shaker
Table Test Coupons," Rohr, Inc. Engineering Report RHR 91-087,
September 1991.
4. R. D. Blevins, "An Approximate Method for Sonic Fatigue Analysis
of Plates and Shells," Journal of Sound and Vibration, Vol. 129,
51-71, 1989.
5. J. W. Miles, "On Structural Fatigue Under Random Loading,"
Journal of Aeronautical Sciences, Vol. 21, November 1954.
944
A. Test Configuration for
Material Coupons
Fixture
Specimen
■ O
B. Test Configuration for
Carbon-Carbon Integral
Stiffener Specimens
Accelerometer
t
C. Test Configuration For
■ Carbon-Carbon Mechanically
Fastened Stiffener and All
Titanium Diffusion Bonded
Joint Specimens
FIGURE 1 Typical Strain Gauge Locations and Test Configurations
for Material Coupon and Joint Subelement Shaker Test
Specimens
945
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950
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SCALING CONCEPTS IN RANDOM ACOUSTIC FATIGUE
BY
Marty Ferman* and Howard Wolfe**
ABSTRACT
Concepts are given for scaling acoustic fatigue predictions for application to
extreme environmental levels based on testing “ scaled” structures at
existing, lesser environmental levels. This approach is based on scaling a test
structure to fit within the capabilities of an existing test facility to attain
fatigue results, and then using analytical extrapolation methods for predicting
the full scale case to achieve accurate design results. The basic idea is to
utilize an existing acoustic fatigue facility to test a structure which has been
designed (scaled) to fatigue within that facility’s limits, employing the
appropriate structural properties ( such as thinning the skins, etc.). Then, the
fatigue life of the actual structure is determined by analytically scaling the test
results to apply to the full scale case ( thicker) at higher noise levels for
example. Examples are given to illustrate the approach with limits suggested,
and with the recognition that more work is needed to broaden the idea.
BACKGROUND
While it is important to continually expand the capability of acoustic test
facilities , it is perhaps equally important to be able to work with existing
facilities at any time. That is, facility expansions, enhancements , and
modernization’s should always be sought from time to time, so long as
practical and affordable from cost effective considerations. Limits should be
pushed to accommodate larger sizes of test specimens with higher noise levels
with wider ranges of frequencies, with wider ranges of temperatures, and with
better capabilities for applying pressures along with any one of several types
of preloads. These are costly considerations and require considerable time to
accomplish. Facility rental can be used in some cases to bolster one’s testing
facilities, however if the application suggests a situation beyond any available
facility for the required design proof, then an alternate is needed. Thus the
scaling concept suggested here is a viable and useable possibility.
The Author’s basis for the approach stems from their extensive, collective,
experience in Structural Dynamics, especially work in Acoustic Fatigue,
Fluid-Structure Interaction, Buffet , and Aeroelasticity/Flutter , and
particularly from experience with flutter model testing, in which it is quite
common to ratio test results from a model size to full scale for valid
Assoc. Prof, Aerospace and Mech. Engr. Dept, Parks College, St. Louis
Univ., Cahokia, IL, 62206, USA
** Aerospace Engineer, Wright Laboratory, Wright Patterson AFB, OH,
45433, USA
953
predictions . Flutter is a well recognized area where model data is commonly
used in nondimensional form to establish design margins of safety, as typified
in Ref (1-2). Flutter can be nondimensionalized quite broadly as pointed out
in many works, and is clearly done for a wide range of general cases using the
“so-called” Simplified Flutter Concept, Ref (3) . The degree of the use of the
flutter model scaling rules varies considerably today, because some people
are testing as much or more than ever, while others are testing less and relying
more heavily on advanced theories such as Computational Fluid Dynamics,
CFD. However, the basic ideas in flutter model scaling are still POWERFUL!
In fact, this concept has fueled the Author’s desires to develop the
“acoustical scaling” used in the approach presented. Moreover, when starting
to write this paper, the Authors realized that this type of scaling is also
common to many related areas of structural dynamics, and thus chose to
include some examples of those areas to emphasize the main point here!
For example, experience in fluid-structure interaction and fatigue of fuel tank
skins, a related work area, serves as another example of scaling structures to
demonstrate accurate predictions with widely varied environmental levels,
and a multitude of configurations. Scaling and nondimensional results were
used extensively in Ref (4 -13), and are cited here because of the immense
data base accumulated. The work at that time did not necessarily define
scaling as used here , but hindsight now suggests that there is a clear relation.
It is becoming well recognized that Buffet is easily scaled , and many
engineers and investigators are now employing scaling of pressures from
model to full size applications, and are also using scaled model response to
predict full scale cases . Some of the earliest and some of the more modern
results clearly show this aspect. For example. Ref (14 -16) are typical, quite
convincing, and pace setting regarding scaled data. Buffet models which are
much more frail that the full-scale cases are used to develop data for full scale
applications, and besides giving full sized results, provide a guide to safe
flight testing as has been done more extensively with flutter testing.
Obviously, acoustical response and fatigue phenomenon are also
nondimensionalizable and scaleable. Ref (17-18), for example. This point is
being taken further here; that is , scaling will be used to take better advantage
of limited facility testing capability to predict more severe situations, as is
used in the case of flutter model testing where a larger specimen is predicted
from tests of a smaller structure using similarity rules. Here in the acoustic
application, a thinner , or otherwise more responsive specimen, is tested and
then analytical means are used to make the prediction for the nominal case.
APPROACH
The method is shown here is basically an extension of the flutter model
scaling idea, as applied to acoustical fatigue testing with a particular emphasis
on random applications. The technique will also work for sine type testing in
954
acoustical fatigue, and perhaps it will be even more accurate there, but most of
today’s applications are with random testing, notably in the aircraft field.
Thus it is in this area where the method should find more application . The
Authors have a combined professional work experience of some 70+ years
and thus have tried to focus this extensive background on an area where gains
can be made to help reduce some costs while making successful designs, by
using lesser testing capability than might be more ideally used. It is believed
that the best testing for random acoustic fatigue, is of course, with (a) the
most highly representative structure, and as large a piece as can be tested,
both practically and economically, (b) the most representative environmental
levels in both spectrum shape and frequency content, (c) test times to
represent true or scaled time, as commonly accepted, (d) temperatures should
be applied both statically and dynamically, and finally (e) preloading from
pressures, vibration, and from boundary loading of adjacent structure.
Frequently, testing is done to accomplish some goal using a portion of these
factors, and the remainder is estimated . Thus the Authors believed that there
is a high potential to extend the flutter model approach to acoustical
applications.
Recall that in the flutter model approach , the full scale flutter speed is
predicted by the rule
((Vf)a)p- [ ((Vi.)m)e/ (( Vf)m)c] X [ ((Vf)a)c] (^)
where Vp is flutter speed, the subscripts M and A refer to model and aircraft
respectively, the subscript C refers to calculated, and the subscript P refers to
predicted. Thus the equation suggests that the full scale predicted flutter speed
is obtained by taking the ratio of experimental to calculated flutter speed for
the model and then multiplying by a calculated speed for the airplane. These
flutter model scaling ideas are covered in any number of References, i.e. Ref
(lo), for example.
The same concept can be utilized in acoustic fatigue, i.e. the strain at fatigue
failure relation, (8,N) can be scaled from model structure tested at one level
and then adjusted for structural sizing and environmental levels. This relation
can be addressed as done for the flutter case:
((e,N)a)p=[(( s,N)m)e / ((e,N)m)c] X [((e.N)a)c] (2)
where s is strain, and N is the number of cycles at failure, where as above in
Eq (1) , the subscripts M and A refer respectively to Model and Full Scale for
parallelism, while the subscripts E, C, and P have the same connotation again,
namely, experimental, calculated, and predicted. Thus the full scale case is
955
predicted from a subscale case by using the ratio of experimental to
theoretical model results as adjusted by a full scale calculation. Flutter model
scaling depends upon matching several nondimensional parameters to allow
the scaling steps to be valid. While these same parameters are, of course, not
necessarily valid for the acoustic relationships, other parameters unique to
this acoustical application must be considered, and will be discussed.
Accurate predictions for the method relies on extensive experience with the
topic of Acoustic Fatigue in general, because concern is usually directed
towards the thinner structure such as; panels, panels and stiffeners, and panels
and frames, bays (a group of panels), or other sub-structure supporting the
panels. These structures are difficult to predict and are quite sensitive to edge
conditions, fastening methods, damping, combination of static and dynamic
loading, and temperature effects. Panel response prediction is difficult , and
the fatigue properties of the basic material in the presence of these complex
loadings is difficult. However, the experienced Acoustic Fatigue Engineer is
aware of the limits, and nonnally accounts for these concerns. Thus the
method here will show that these same concerns can be accounted for with the
scaling approach through careful considerations.
The Authors believe that the method is best explained by reviewing the
standard approach to acoustic fatigue, especially when facility limits are of
major concern. Fig (1) was prepared to illustrate these points of that
approach. Here it is seen that key panels for detail design are selected from a
configuration where the combination of the largest, thinnest, and most
severely loaded panels at the worst temperature extremes and exposure times
are considered. These can be selected by many means ranging from empirical
methods, computational means, and the various Government guides. Ref (17-
18), for example. Then detailed vibration studies are run using Finite
elements , Rayleigh methods. Finite Difference methods, etc. to determine the
modal frequencies and shapes, and frequently linearity is assessed. Then
acoustical strain response of the structure is determined for sine, narrowband,
and broadband random input to assess fatigue life based on environmental
exposure times in an aircraft lifetime of usage. These theoretical studies are
then followed by tests of the worst cases, where vibration tests are conducted
to verify modal frequencies, shapes, and damping, and linearity is checked
again for the principal modes. This is followed by acoustical strain response
tests where the strain growth versus noise levels is checked, again employing
sine, narrowband and broadband random excitation. Note the figure suggests
that data from the vibration tests are fed back to the theoretical arena where
measured data are used to update studies and to correlate with predictions,
especially the effect of damping on response and fatigue, and of course, the
representation of nonlinearity. Also, the measured strain response is again
used to update fatigue predictions. These updates to theory are made before
the fatigue tests are run to insure that nothing is missed. However, in this
956
case, the required sound presssure level SPL in (dB) is assumed to exceed the
test chamber’s capability. Thus , as shown in the sketch in Fig (2) the key
strain response curve, s vs dB, is extrapolated to the required dB level. This
data is merged with the strain-to-failure curve at the right to establish the cycle
count, N, giving the (s, N) point for this case. The extrapolated data provides
some measure of the estimated life, but again is heavily dependent upon the
accuracy of the basic strain response curve, and is especially dependent on
whether high confidence exists at the higher strains. Linear theory is also
shown in this case, indicating it overpredicts the test strain response and hence
shows a shortened fatigue life compared to test data, as is generally the case
in today’s extreme noise levels. This illustration is highly simplified,
because experienced designers readily know that it is difficult to predict even
simple panels accurately at all times, let alone complex and built-up structure
consisting of bays (multi-panel); this will addressed again later in the paper.
The new concept of scaled acoustic fatigue structures is shown on the sketch
of Fig (3) where the standard method is again shown, but smaller in size, to
refocus the thrust of the new idea. The scaling process parallels the standard
approach, and actually complements that approach, so that the two can be run
simultaneously to save time, costs, and manpower. Here the panel selection
process recognizes that the design application requires environments far in
excess of available facility capability. Thus the scaling is invoked in the
beginning of the design cycle. As the nominal panels (bays) are selected and
analyzed for vibration, response and fatigue, scaled structures are defined to
provide better response within the existing chamber ranges so that they can be
fatigued and then the results can be rescaled to the nominal case. In this
manner, appropriate designs can be established to meet safety margins with
more confidence, and will avoid costly redesign and retrofitting at
downstream stages where added costs can occur and where down times are
difficult to tolerate. The concept is further illustrated in the sketch of Fig (4).
Here the strain response curve of the nominal case and that of the scaled
version are combined with strain to failure data ( coupon tests) to show fatigue
results. Note the strain response for the nominal case at the highest dB level
available gives the fatigue value at point A, while the extrapolated data for
this curve gives point B. The scaled model being more responsive gives the
point C, and when rescaled gives the point D which differs slightly from the
extrapolated point B as it most likely will, realistically. More faith should be
placed on data from an actual fatigue point than a point based on the projected
strain response curve. Note, Fig (5) illustrates the winning virtue of the
scaleable design. The figure shows a hypothetical set of test data for the (s ,N)
for a structure for various SPLs for the nominal case, open circles, and for the
fatigue results of the scaled model , closed squares. The scaled model was
assumed to be thinner here for example, and that the scaled data is also
957
rescaled to fit the nominal curve here. The most interesting aspect is shown
by the two clusters of data , denoted as A and B where there are rough circles
about drawn about the clusters. Here the emphasis is that tests of the scaled
model ( and rescaled data ) are used to find the higher strain conditions which
cannot be found from the nominal case. In both cases at the highest strain
levels, the facility is used to its limits, but with enough testing with the thinner
case, adequate data is available to make the prediction more accurate using
Eq (2) for the final correlation as shown here. The statistical scattering of the
scaled data will be an accurate measure for the nominal case, particularly
when compared to estimates based on extrapolation of the strain response for
the nominal case. There are many cautions to be noted with this approach as
there are with all acoustic fatigue methods, and of course, tests. First, the
linearity of the modes, either in unimodal sine excitation, multi-mode sine,
narrow band or broadband random must be carefully handled. The strain
response of individual locations throughout the structure must be carefully
monitored in calculations and tests so that strain response is truly understood
and used to define fatigue life carefully. This is difficult to do in many
applications where widely varying conditions and durations require some type
of Miner Rule combination to provide a true measure of fatigue. Similarly,
strain risers at fasteners, discontinuities, holes, frames, stiffeners, material
changes along with temperature gradients, temperature transients, require
final “tweeks” to predictions, regardless. Nonlinearity , especially in the
multi -mode case, is one of the most formidable foes to conquer for any
application.
APPLICATIONS AND EXAMPLES
The tests of an Aluminum panel of size 10x20 in. and with a thickness of
0.063 thickness , Ref (19), will be used to illustrate the technique. The panel
has approximately fixed-fixed edge conditions, and is quite nonlinear in
terms of strain response. Ref (19). The measured strain response for the panel
is shown in Fig (6) along with an estimated response curve for a thinner panel
( 0.040 in ) based on the test data. For this case it is assumed that data were
needed at 175 dB, while the facility could only achieve 164 dB. The strain
response for the thinner case was estimated using the classical equation for
the amplitude of response, 5 ;
\\fdxdy\pSD,{f)]
(3)
where M is the generalized mass, co is the natural frequency, § is the viscous
damping factor , (j) is mode shape, PSDp is the pressure Power Spectral
Density, and x,y are the positional coordinates along the plate. Since strain ,
s, is proportional to the amplitude.
958
s= (t/2)(3(ti^/ax^)S
(4)
Combining Eq (3 and 4) shows that the strain response curves are
proportional to the thickness factor, as given by:
s 2 ~ ( ti / t2) (5)
However, it must be noted that this case is nonlinear, and thus, this result is
not exactly correct, but simply used for an illustration here. The actual data for
the 0.063 thickness is extrapolated to a required 175 dB, showing a strain of
1000 micro in/in. The estimated curve for the thickness of 0.040 in. shows ,of
course, a greater response at all dB levels as it should, and moreover shows
that only 150 dB are needed to achieve the 1000 micro - strain condition.
Moreover, the thinner panel will exhibit large enough strains at the lower
SPLs to improve the fatigue curve where the thicker panel is insensitive.
Taking the example a step further, the fatigue point of the nominal case is
shov^Ti on a strain to failure plot in Fig (7) , employing beam coupon tests of
Ref (11), which were shown to be excellent correlators with panel fatigue in
the collection of work in Ref (6-13). The fatigue point for the 164 dB
excitation, 800 micro-strain, is shown as a triangle, while the extrapolated
data for 175 dB is shown as the flagged triangle. One test point exists for the
nominal case. Ref (20) , and is shown by a star symbol. Data for the thinner
panel are shown as circles at the various strain to cycle count cases for the
various SPLs corresponding to the beam curve. Notably, these points can be
seen to produce shorter fatigue cases as they should due to increased strains,
but note that they are also at lower frequencies which would give a longer test
time than if they were the nominal thickness. The scaled model is seen to
produce the same point as the extrapolated case in this hypothetical case for
the 1000 microstrain case ( again, at two different dB levels for two
thicknesses). A SPL of 150 dB, rather benign, is seen to be quite effective.
The actual fatigue point at 164 dB for the nominal case required 3 hours and
was predicted to be 2.8 hrs. The estimated fatigue for the extrapolated case of
175 dB was estimated to be 1.7 hrs, while the scaled point from the thinner
panel was estimated to be 2.2 hours which is slightly off, but the Authors
have had to rely on log plots for much of the data and thus lack someaccuracy.
Because of lack of actual data , the scatter from the estimate vs the test of the
nominal case was used to scatter the estimate for the 1000 microstrain case,
flagged dark circle, as if the use of Eq (2) had been employed directly. One
must be careful here, because there can be a vast difference between theory
and test, and this can mislead inexperienced persons applying these methods.
As noted earlier, related work in fluid-structure and buffet , actually
demonstrate this type of scaling. To illustrate, several figures are republished
959
here to make this point rather clear. Fig (8) of Ref (8 ) shows the dynamic
bending strains in the bottom panel of an otherwise rigid fuel tank which is
being excited vertically with moving base input. The vertical axis is strain
while the abscissa is the number of g’s input. Three panel thicknesses and four
depths of fluid (water in this case) were used. Note the sharp nonlinear effect
in the response, rather than linear response growth as force increases.
Interestingly, the data was nondimensionalized into the curve of Fig (9), Ref
(8), which was originally intended for a design chart to aid in developing
strain response characteristics for use in fatigue. This curve displays a
parameter of response as the ordinate vs an excitation parameter on the
abscissa. Here, E is Young’s modulus, p is density, t is thickness, a is the
panel length of the short side, h is fluid depth, and the subscripts, p and F refer
to panel and fluid, respectively. A point not realized previously is that the
scaling shows that the thinner case can be used to represent the thicker panel
under the appropriate conditions and when nonlinearity is carefully
considered. More data with the thinner panels at the extreme conditions were
unfortunately not taken in several cases of strain response because of concern
with accumulating too many cycles before running the actual fatigue tests;
else the thinner cases could have shown even more dramatically the scale
effect.
Buffet has been of more interest in the past 15 years because of high angle of
attack operation of several modem USAF fighters. Much effort was placed
upon research with accurately scaled models to detennine if these could be
employed as in prior flutter work. The answer was YES! Several figures were
taken from Ref (16) to illustrate scaling of data from a model of , a fraction
of the size of a fighter, to the full scale quantity. Fig (10) shows the correlation
between scaled-up model data, flight test, and two sets of calculations over a
wide range of aircraft angle of attack for the F/A-18 stabilator. The data is for
inboard bending and torsion moment coefficients produced by buffeting
loads. The scaled model data correlates well, the calculations using Doublet
Lattice (DLM) aerodynamics is close, while the strip theory is not as accurate.
Fig (11) shows similar type of data for the F/A-18 Vertical Tail for outboard
bending moment coefficients. Here a wider range of angle of attack was
considered, and again scaled model data and calculations are close to aircraft
values. Both cases suggest that model data can be used to supplement full
scale work and that when combined with theory , are a powerful aid to full
scale analysis and tests. These tests can be used early in the aircraft design
cycle to insure full scale success.
CONCLUSIONS AND RECOMMENDATIONS
An attempt was made to employ a view of acoustical scaling different from
that usually taken. The idea is to develop data for a model that fits within a
test facility’s capability and then by using analytical methods, adjust these
results to the nominal case using factors from the test based on the ratio of
960
experimental to calculated data. This is analogous to the flutter model
approach. One example is offered, and similar results from related scaling in
fluid -structure and buffet work were shown to further the point. While more
work is needed to fully display the concept, enough has been done to inspire
others to dig-in and more fully evaluate the approach. The Writers intend to
do more research, since they fully appreciate this difficult task.
REFERENCES
1. Scanlon , R.H., and Rosenbaum, R., “ Introduction to The Study of Aircraft
Vibration and Flutter”, The MacMillian Company, New York, 1951
2. Bisplinghoff, R.L., Ashley , H. and , Halfman, R. L., “Aeroelasticity”,
Addision-Wesley Publishing Co., NY, Nov. 1955, pp. 695-787
3. Ferman, M. A., “Conceptual Flutter Analysis Techniques - Final Report”
Navy BuWeps Contract NO w 64-0298-c, McDonnell Report F322, 10 Feb.
1967
4. Ferman, M.A. and Unger, W. H., “Fluid-Structure Interaction Dynamics in
Fuel Cells”, 17th Aerospace Sciences Meeting, New Orleans, La. Jan 1979
5. Ferman, M.A. and Unger, W. H. , “Fluid-Structure Interaction Dynamics in
Aircraft Fuel Cells”, AlAA Journal of Aircraft, Dec. 1979
6. Ferman, M.A. , et al , “ Fuel Tank Durability with Fluid-Structure
Interaction Dynamics ,” USAF AFWAL TR-83-3066, Sept. 1982
7. Ferman, M..A., Unger, W. H., Saff, C.R., and Richardson, M.D., “A New
Approach to Durability Predictions For Fuel Tank Skins” , 26th SDM,
Orlando, FL, 15-17 April 1985
8. Ferman, M. A. , Unger, W. H., Saff, C.R., and Richardson, M.D. , “ A
New Approach to Durability Prediction For Fuel Tank Skins”, Journal of
Aircraft, Vol 23, No. 5, May 1986
9. Saff, C.R., and Ferman, M.A, “Fatigue Life Analysis of Fuel Tank Skins
Under Combined Loads”, ASTM Symposium of Fracture Mechanics ,
Charleston, SC, 21 March 1985
10. Ferman, M.A., Healey, M.D., Unger, W.H., and Richardson, M.D.,
“Durability Prediction of Parallel Fuel Tank Skins with Fluid-Structure
Interaction Dynamics”, 27th SDM, San Antonio, TX, 19-21 May 1986
11. Ferman, M.A, and Healey, M.D., “Analysis of Fuel Tank Dynamics for
Complex Configurations, AFWAL TR -87-3066, Wright-Patterson AFB, OH,
Nov 1987
12. Ferman, M.A., Healey, M.D. and Richardson, M.D.,” Durability Prediction
of Complex Panels With Fluid-Structure Interaction”, 29th SDM,
Williamsburg, VA, 18-20 April 1988
13. Ferman, M.A., Healey, M.D., and Richardson, M.D., “A Dynamicisf s
View of Fuel Tank Skin Durability, AGARD/NATO 68th SMP, Ottawa,
Canada, 23-28 April 1989
14. Zimmerman, N.H. and Ferman, M.A., “Prediction of Tail Buffet Loads
for Design Applications, USN Report, NADC 88043-30, July 1987
961
15. Zimmerman, N.H., Ferman, M.A., Yurkovich, R.N, “Prediction of Tail
Buffet Loads For Design Applications”, 30th SDM, Mobil, AL , 3-5 April
1989
16. Ferman, M.A., Patel, S.R., Zimmemian, N.H., and Gerstemkom, G., “ A
Unified Approach To Buffet Response Response of Fighters”, AGARD/NATO
70th SMP, Sorrento, Italy, 2-4 April 1990
17. Cote, M.J. et al, “Structural Design for Acoustic Fatigue”, USAF ASD-
TR-63-820, Oct 1963
18. Rudder, F.F., and Plumblee, H.E., Sonic Fatigue Guide for Military
Aircraft” AFFDL-TR-74-1 12, Wright-Patterson AFB, OH, May 1975
19. Jacobs, J.H., and Ferman, M.A. , Acoustic Fatigue Characteristics of
Advanced Materials and Structures, “ AGARD/NATO SMP , Lillehammer,
Norway, 4-6 May 1994
20. McDonnell Douglas Lab Report, Tech. Memo 253.4415, Acoustic Fatigue
Tests of Four Aluminum Panels, Two With Polyurethene Sprayon”, 27 June
1984
Figure 1 - Standard Acoustic Fatigue Design Method
962
MICROSTRAIN -
HYPOTHETICAL EXAMPLE
SOUND PRESSURE LEVEL -dB N- CYCLES TO FAILURE
Figure 2 - General Method for Estimating Fatigue Life at SPL
above Test Facility Capability
Figure 3 - Scaling Method Fits-in with General Design
Cycle for Acoustic Fatigue
963
SOUND PRESSURE LEVEL - dB
N@ cIBreq , Nom. Ext.
N @ dBj^ , Re-scaled
N @ dBj^ , Scaled Model
N - CYCLES TO FAILURE
Figure 4 - Hypothetical Examples of Acoustic Scaling
to Tests at Higher SPLs
HYPOTHETICAL CASE
Figure 5 - Statistical Aspects of Scaling
MICROSTRAIN -
^ MDC Tests
Ref. (19)
Figure 6 - Strain Response of an Aluminum Panel
(10 X 20 X 0.063 in. 7075 T6) Narrowband Random
Figure 7 - Example of scaled Model of the 10 x 20 in.
Aluminum Panel
965
Symbol Panel Fluid
Thickness Depth
(in.) (in.)
□
0.032
11.0
0
0.040
11.0
o
0.063
11.0
0
0.032
8.0
A
0.040
8.0
o
0.063
8.0
0
0.032
4.0
0.040
4.0
•
0.063
4.0
Figure 8 - Dynamic Strain vs Excitation Level - Sine
Figure 9 - Dynamic Strain Parameter vs Input Parameter
Mean Strain at Fluid Depth
966
0.04
Inboard Bonding
Inboard Torsion
0.032
RMS Buffet 0.024.
Moment
Coeffidant,
0.01 e
0.008
0 .
0 4 8 12 16 20 24
Angle of Attack - degrees
A
C
1
1
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1 0
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0 4 8 12 16 20 24
Angie of Attack - degrees
Figure 10 - F/A-18 Stabilator Buffet Correlation Study
0.048
0.04
0.032
RMS
Moment
CoefficiantO.024
BM/(QL3)
0.016
0.008.
0
16 24 32 40 48 56
Angle of Attack - degrees
Figure 11 - F- 18 Vertical Tail Buffet Response Moment Coefficients for
Angle-of-attack Variations
Outboard Bending Moment (70% Span, 45% Chord)
j— I j j
—
□ Ca
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968
ACOUSTIC FATIGUE II
THE DEVELOPMENT AND EVALUATION OF A NEW
MULTIMODAL ACOUSTIC FATIGUE DAMAGE MODEL
Howard R Wolfe
WL/FIBG Bidg24C
2145 Fifth St Ste2
Astronautics
Wright-Patterson AFB, OH
45433-7006
USA
Robert G. White
Head of Department
Department of Aeronautics and
University of Southampton
Southampton, S017 IBJ
UK
ABSTRACT
A multimodal fatigue model has been developed for flat beams and plates.
The model was compared with experimental bending resonant fatigue
lifetimes under random loading. The method was accurate in predicting
cantilevered beam fatigue lifetimes, but under predicted clamped- clamped
beam test results. For the clamped plate tests, one calculation was accurate
and the other predicted about half the test lifetimes. The comparisons and
the parameters affecting them are presented.
INTRODUCTION
While the single mode acoustic fatigue theory is satisfactory for sound
pressure levels around 158 dB overall and below, there is evidence in the
literature [1, 2, and 3] that above this level the accuracy of the simple
response prediction method decreases with increasing sound pressure
levels. The purpose of the paper is two fold, first to develop a multimodal
acoustic fatigue life prediction model ^d secondly, to evaluate its accuracy
in estimating the fatigue life theoretically by comparing predictions with
experimental results.
FATIGUE MODEL DEVELOPMENT
Many fatigue models are found in the literature. The Miner single mode
model used by Bennouna and White [4] and Rudder and Plumblee [5] was
selected to develop a multimodal nonlinear model. The fundamental
formulation is given by,
N.-fs^r «)
969
where N,. is the total number of cycles to failure, P(e) is the peak strain
probability density. N is the total number of cycles to failure at incremental
constant amplitude strain levels derived from a sinusoidal strain versus
cycles to failure curve. To calculate the fatigue life in hour, Eq (1) can be
expressed as,
t (hours) =
where t is time, Pp (sd) is the peak standard deviation probability density,
Nc is the total number of cycles to failure at a specified strain level and ^ is
the cyclic frequency. When the mean value is not zero, which is the case
with axial strain in the beam or plate, the rms value is not the standard
deviation. The standard deviation is usually employed to compute the time
to failure. Most of the S-N curves or e -N curves are approximated as a
straight line on a logarithmic graph. The relationship between the surface
strain and the cycles to failure is then.
8 =
(3)
where K is a constant and a the slope of a straight line on a log-log graph.
The cyclic frequency 4 for a single mode case is taken as the frequency of
the associated resonance. Two types of peak probability density techniques
were investigated from multimodal nonlinear strain responses [3]. These
were called major peaks and minor peaks. The major peaks were counted
for the largest peaks between zero crossings. The minor peaks were
counted for all stress reversals or a positive slope in the time history
followed by any negative slope. The effective cyclic frequency is much
lower for the major peak count than the minor peak count. However, the
peak probability density functions or PPDFs compared for these two cases
were almost the same. The major peak method was selected for further
study.
Given a particular peak probability density curve from a measured
response in an experiment, the number of peaks and the sampling time tg
can be used to determine the effective multimodal cyclic frequency,
fcm = number of peaks / tg (4)
970
where is the effective multimodal cyclic frequency. Substituting the
multimodal cyclic frequency into Eq (4),
t (hours) =
Pp(sd)
K/(e)‘/“
3600xfcm
(5)
This model accounts for the effects of axial strains which cause the mean
value not to be zero, nonlinear response and multimodal effects. If the
mean value is zero, then the standard deviation is equal to the rms value
and Eq (5) reduces to Eq (2).
FATIGUE MODEL COMPARISON WITH EXPERIMENTAL
RESULTS
The peak probability density function (PPDF) is needed or preferably
the time history from strain or dynamic response measurements to
evaluate the fatigue model developed. Also needed are sinusoidal 8 -N
curves for the structure, and knowledge of its boundary conditions and the
equivalent multimodal cyclic frequency.
Two types of peak probability density techniques were investigated from
multimodal nonlinear strain responses [3]. These were called major peaks
and minor peaks. The major peaks were counted for the largest peaks
between zero crossings. The nunor peaks were counted for all stress
reversals or a positive slope in the time history followed by any negative
slope. The effective cyclic frequency is much lower for the major peak coimt
than the nunor peak coimt However, the peak probability density
functions or PPDFs compared for these two cases were almost the same.
The major peak method was selected for further study.
Comparison with Beam Data:
The K and a terms were calculated from 8-N and S-N data, where S
is stress, using Eq (3). Selecting two values of strain and their
corresponding cycles to failure, yields two simultaneous equations
which were solved for K and a. Two sinusoidal E-N curves for
BS1470-NS3 aluminum alloy which has a relatively low tensile strength
were obtained from Bennouna and White [4 Fig 8]. These were for a
cantilevered beam and a clamped-clamped (C-C) beam as shown in Fig
1. The K and a terms calculated were used to compute the cycles to
failure, Nc, for each strain level. Table I shown is the same as Table I in
reference [4] except was calculated from Eq (3) to sum the damages.
971
Delta is the sample size. The cyclic frequency was for one mode the one
resonant response frequency. The time to failure in hours using Eq (2)
for the cantilevered beam was 16.6 hours compared with 16.2 predicted
theoretically [4], 15.3 and 15.9 obtained experimentally [4]. Both
theoretical results were essentially equal, but slightly higher than the
test results. The time to failure from Eq (2) for the (C-C) beam was 3.04
hours compared with 2.53 theoretically [4] and 5.25 and 5.92
experimentdly [4]. Both theoretical results were about one half of the
test results. The K and a terms, the theoretical fatigue life times and the
experimental fatigue life times are listed in Table H. The table contains
three sections: lifetimes calculated using a strain gauge PPDF, a
displacement PPDF and the Gaussian and Rayleigh PDFs. As noted in
reference [4], failure occurred much earlier for the C-C beam than the
cantilevered beam for the same strain level. This was attributed to the
influence of a large axial strain in the clamped- clamped beam.
Comparison with Plate Data:
Two fatigue tests were conducted to provide some additional limited
data for comparison with the fatigue model developed. These tests
used the base excitation method with a 1.09x10® N (20,000 Ibf)
electrod)mamic shaker. The clamping fixture consisted of a flat
aluminum alloy 6061-T6 plate 19 mm thick and four clamping bars of
equal thickness. The radius of curvature of the clamping edges was 4.76
mm to prevent early fatigue failure. A four bar clamping arrangement
was selected to prevent buckling of the plate while torquing the
clamping bolts. The undamped size was 254 x 203 x 1.30 mm which
results in a 1.25 aspect ratio. Strain gauges were bonded along the center of
the larger dimension (SG 2) and at the center of the plate (SG 3).
Displacements were measured with a scanning laser vibrometer at the
center of the plate. An accelerometer was moimted on the shaker head to
determine the acceleration imparted to the damped plate. A flat
acceleration spectral density was used between 100-1500 Hz. Recordings
were taken at increasing levels of exdtation up to the fatigue test level. The
time to detecting the first fatigue crack was recorded for each plate.
The constants K and a were calculated from random single mode S-N
data for 7075-T6 aluminum alloy [5 p 489] shown in Fig 1, with
K = 1.01x10^^ and a = - 0.175. The rms stress ( Srms ) ^^s changed to rms
strain, Srms = EBrms, where E is Young's modulus. The stress was
measured half way between two rivets along the center line between the
rivets on the test specimen. The strain gauge location, stress
concentrations, and the boundary conditions greatly affect the strain
972
level measured. Correction factors are needed for a different set of
conditions and to convert random data to sinusoidal data. Sinusoidal
£ -N bending coupon curves for 7075-T6 aluminum alloy were difficult to
find. S-N curves were found for an aerospace material with both
sinusoidal and random excitations. These curves were nearly parallel.
The sinusoidal strain was 1.38 times larger than the random strain for
10^ cycles. Multiplying the constant K for the 7075-T6 material by 1.38
resulted in K = 1.40x10^^ .
Early strain gauge failures prevented strain measurement above 500
microstrain with 20.7 g rms shaker excitation. The fatigue test level was
115 g rms and the response contained at least six frequency response
peaks. The major peak strain PPDFs were determined for 5.32 g rms
and 20.7 g rms as shown in Fig 2 with the Gaussian PDF. Compared
with the Gaussian distribution, an increased number of peaks occurred
greater than 1 sigma and smaller than -1 sigma. Also a larger number of
peaks occurred around zero. The PPDF determined from the 20.7g rms
test case was used to predict fatigue life, but a new strain estimate was
needed since the excitation level increased 5.6 times. The displacement
is directly related to the strain at each excitation level. Displacement
measurements at the fatigue test level were used to estimate the strain
level shown in Fig 3. The estimated strain from the figure was 770
microstrain for SG 2. The scale of the displacement measurements was
adjusted to coincide with strain measurements at increasing increments
of shaker excitation.
The equivalent cyclic multimodal frequency is needed to predict the
fatigue life. Prediction of the linear modal frequencies is carried out by
a variety of methods. Usually the first mode prediction is the most
accurate. The cyclic multimodal nonlinear frequencies have been
studied for two clamped beams and two clamped plates [3]. These were
based upon the peak probability density functions (PPDFs) where the
peaks were counted for a specific time interval, from which the
nonlinear cyclic multimodal frequencies were calculated. Generally the
resonant frequencies increased with increasing excitation levels. Those
for the two beams increased more rapidly than those for the two plates.
Very little change was noted for the plates. The equivalent cyclic
miiltimodal frequency determined via Eq (4) from the SG 2 PPDF was
348 Hz.
The time to failure in hours predicted using Eq (5) and the parameter
mentioned above for the clamped plate was 0.706 hours compared with
1.17 and 0.92 shown in Table 11. The predicted result was slightly lower
973
than the test results. A Srms’^ curve was calculated with the test
lifetimes available, by determining a new constant K, assuming the
slope was the same as for the riveted coupon and applying the
sinusoidal correction factor. The time to failure in hours predicted
using Eq 5 and the calculated e-N curve for the clamped plate was
0.274 hours compared with 1.17 and 0.92 shown in Table E. The cyclic
multimodal frequency used was the same at that determined from SG 2
and the same strain was used. This prediction was about 1/3 of the test
results. This method incorporates the failure data at two points.
The displacement PPDF shown in Fig 4 was used to predict fatigue
life. The number of displacement peaks increased significantly above
the strain PPDF around 1 sigma and -1 sigma. The large number of
peaks around zero was similar to the characteristics observed in the
strain PPDF. The equivalent cyclic multimodal frequency was 375 Hz,
slightly higher than that determined from the strain PPDF. However,
the same frequency (348 Hz) was used to predict fatigue life. The time to
failure in hours using Eq (5) for the clamped plate was 1.15 hours with
the riveted e-N curve and 0.446 with the calculated 8-N curve shown
in Table E. This PPDF improved considerably the prediction. The
Gaussian and Rayleigh PDFs were used to predict the time to failure
with the same parameters as those used with the riveted e-N curve.
The lifetime using the displacement PPDF was 1.15 hours, using the
Gaussian PDF, 0.600 hours and using the Rayleigh PDF, 0.237 hours, as
shown in Table E. The Gaussian PDF under predicts by a factor of 2.
The Rayleigh PDF under predicts by a factor of 5.
The spread sheets containing PPDF / Ncm data for various sigma
values were used to determine damage accumulation shown in Fig 5.
Almost 55% of the normalized damage occurs between -2 and -1 sigma
and 38% of the damage between 1 and 2 sigma using the displacement
PPDF. However, the damage is spread more evenly using the available
strain gauge PPDF. The strain gauge PPDF was recorded at a much
lower level than the displacement PPDF. The damage accumulation
compared more closely to Gaussian PDF than the Rayleigh function.
Damage Model with a Specific Function Describing the PPDF:
A curve-fitting routine was used to determine a mathematical function
for a high level strain gauge PPDF for the clamped shaker plate. The
most important part of the fit is outside the range of -1 to 1 sigma, since
most of the damage accumulation occurs outside this range. The
highest ranking function was a tenth order polynomial followed by
974
ninth and eighth order polynomial fits. The goodness of fit in order
from 1 to 14 ranges from 0.9775 to 0.9625, which are very close
statistically. The tenth order polynonual is,
y = a 4* bx + cx^ + dx^ + ex"^ + fx^ + gx^ + hx^ + ix^ + jx^ + kx^^ (6)
where a=0.346, b=-0.0148, c=-0.137, d=-0.054, e=0.090, f=0.043, g=-
0.0400, h=-0.00976, i= 0.00722, j=6.85xl0"‘, k=<4.36xl0'" . Ranked
fourteenth is a natural logarithmic function. The function and its
coefficients are,
Iny = a + bx + cx^ + dx^ +ex^ 4-fx^ (7)
where a=-1.088, b=-0.1191, c=-0.1.302, d=0.0104, e=-0.0653, and f=0.0079.
The function fits the test data similar to the tenth order polynomial and
may be easier to use. Ranked forty-first is a Gaussian function. The
function and its coefficients are.
y = a + b exp|o.5[(x - c) / d]^ |
(8)
where a=-0.0968, b=0.4485, c=-0.050 and d=1.45. The function fits better
for sigma values of 2 or greater than those of -2 sigma and greater. A
constant coefficient is used to fit the Gaussian function to permit shifting
the function to fit the test data. This equation can be used in the PPDF
in Eq (5),
a + b exp [-0.5 [(x - c) / d]^ }
- [K/(e)f“
3600£
■cm
(9)
where a=-0.0968, b=0.4485, c=-0.050 and d=1.45.
CONCLUSIONS
The prediction of multimodal fatigue life is primarily dependent upon
the peak probability density function (PPDF) which changes shape with
increasing excitation levels. The next in order of importance is the
sinusoidal e -N bending fatigue curve and finally the effective multimodal
cydic frequency.
A multimodal fatigue model was developed with the PPDF estimated
from a form of the Gaussian function being useful especially in the
975
range of cydes to failure from 10^-10^. The lifetime predication
calailations for the damped-clamed beam was about one half the
experimental value. For the plate, the calculations was about one half
the experimental value. Using riveted coupon fatigue data, the
calculation was accurate.
REFERENCES
1. B.L. Clarkson, April 1994, '‘Review of sonic fatigue technolog/',
NASA contractor report 4587, NASA Langley Research Center,
Hampton, Virginia.
2. R.G. White, October 1978, "A comparison of some statistical
properties of the responses of aluminium alloy and CFRP plates to
acoustic excitation". Composites 9(4), 125-258.
3. H.F. Wolfe, October 1995, "An experimental investigation of
nonlinear behaviour of beams and plates excited to high levels of
dynamic response", PhD Thesis, University of Southampton.
4. M. M. Bennouna, and R. G. White, 1984, "The effects of large
vibration amplitudes on the dynamic strain response of a clamped-
clamped beam with consideration on fatigue life". Journal of Sound and
Vibration, 96 (3), 281-308.
5. J. R. Ballentine, F. F. Rudder, J. T. Mathis and H.E. Plumblee,
1968, 'Refinement of sonic fatigue structural design criteria", AFFDL TR
67-156, AD831118, Wright-Patterson AFB, Ohio.
TABLE I
FATIGUE CALCULATIONS USING EQUATION 5.9
CANTILEVERED BEAM BS 1470-NS3 (REF 4 FIG 8)
e/sd
sd
P£
e
ll£
Nc=(2.172
xl0“/sd)“““
PPDF X A
PPDF X A/Nc
t
hours
0.5
H
213
15079045.41
n
1.41256E-07
1
|g
425
4390936.857
mam
7.06000E-07
1.5
425
638
2125446.064
WBrn
1.12917E-06
2
425
850
1273261.536
0.138
1.08383E-06
2.5
425
1063
854026.6252
0.060
7.02554E-07
3
425
1275
617189.6449
0.018
2.91645E-07
3.5
TOTAL
425
1488
468372.0662
0.005
0.984
1.06753E-07
4.16121E-06
16.7
976
TABLE n
SUMMARY OF FATIGUE CALCULATIONS
Figure 1 8 -N curves for aluminum alloys tested.
Figure 2 Normalized strain PPDF comparison with a Gaussian PDF.
978
NORMALIZED DISPLACEMENT
MOJORPPDF
2
Figure 4 Normalized displacement PPDF comparisons,
9
980
ACOUSTIC FATIGUE AND DAMPING TECHNOLOGY IN
COMPOSITE MATERIALS
By B. Benchekchou and R.G. White
Abstract
Considerable interest is being shown in the use of composite materials in
aerospace structures. Important areas include development of a stiff, lightweight
composite material with a highly damped, high temperature polymer matrix
material. The study described in this paper concerns the application of such
material in the form used in thin skin panels of aircraft and investigation of its
fatigue properties at room and high temperature. For this purpose, flexural fatigue
tests have been carried out at two different temperatures and harmonic
three-dimensional FE analyses were performed in order to understand the
dynamic behaviour of plates. Random acoustic excitation tests using a
progressive wave tube, up to an overall sound pressure level of 162 dB, at room
temperature and high temperatures were also performed in order to investigate the
dynamic behaviour of panels made of the materials. Various methods for
including damping in the structure were examined when parameter studies were
carried out, and conclusions have been drawn concerning optimal incorporation
of a highly damped matrix material into a high performance structure.
1-Introduction
Significant areas of primary and secondary structures in military aircraft operate
at high temperature and are subjected to high levels of random acoustic loading,
because of their closeness to jet effluxes. There is then a need to develop a carbon
fibre reinforced plastic material with a high temperature polymer matrix and high
fatigue resistance. Highly damped composite structures should be developed in
order to better resist dynamic loading and to have an enhanced fatigue life. Work
previously carried out on improving the damping in fibre reinforced plastic (FRP)
composites as well as the number of approaches which can be taken to improve
the damping properties of polymeric composites have been summarised in [1].
The aim of the research described here was to study lightweight composite
materials with a highly damped, high temperature polymer matrix material, by
981
investigating its mechanical and acoustic fatigue properties, the latter investigation
being carried out using thin, multilayered plates.
2-ExperimentaI work
For this type of study, two adequate prepregs were highlighted after investigation:
SE300 and PMR15. The SE300 material was carbon fibre reinforced prepreg of
(0°/90°) woven form, 0.25 mm thick and had 60% fibre volume fraction, with no
suitable data available on the material properties. Dynamic mechanical thermal
analyser (DMTA) analyses carried out on four specimens ( 20 nun long and 12
mm wide) with different lay-ups i.e. (0°/90°)4, (+45°/-45°)4, (0°/90°, 45°/45°)s
and (45®/45°;0®/90°)s, allowed to get provided the material properties. Results
from DMTA analyses are shown in Table l.a where the loss factor and the
Young’s modulus values at 40^C and at the glass transition temperature Tg are
presented. The loss factor values varied from 0.0097 to 0.085 for a range of
temperature from 40®C to 300®C.
The PMR15 prepreg was also of (0^/90^) woven carbon form and had 58%
volume fraction. Six DMTA specimens having the following lay-ups: (0®/90®)4,
(00/90^)8, (4450/-450)4, (+450/-45<^)8, (0^/90^-, 450/450)s and
(450/450.00/900)^ were made. Results from DMTA analyses showed that the loss
factor values varied from 0.0129 to 0.0857 for a range of temperature from 40®C
to 400®C, with a value of loss factor of 0.1293 at 375®C, the maximum
temperature for normal use being 352^C. The loss factor and Young’s modulus
values at 40®C and Tg are given in Table l.b.
Mechanical behaviour of the selected materials
The fatigue characteristics of these new materials were investigated and results
were compared with those of well established structural materials. Mechanical
fatigue tests of SE300 and PMR15 samples using "sinusoidal" loading at a chosen
maximum strain level, i.e. 8000 pS were carried out and performances compared
to that of an XAS/914 sample. A mechanical (flexural) fatigue rig was used for
this purpose to test specimens in a cantilevered configuration. Details of the rig are
available in [2]. The particular clamp used was designed by Drew [3] to induce
damage in the centre of the specimen instead of having edge damage, i.e. peeling
while flexural tests are carried out. In order to investigate the performance of these
new materials at high temperature, fatigue tests were also carried out on samples,
at 210^C . This was achieved by using a heating system which consisted of two air
982
blowers (electronically regulated hot-air guns) positioned at 40 mm above and
below the specimens, which allowed specimens to be tested at a uniform temperature
of 210±5°C. The aim of the mechanical fatigue test was to determine the number of
loading cycles needed for damage to occur and its subsequent growth rate in
cyclically loaded composite specimens of SE300 and PMR15 matrix materials . The
samples were 140 mm long, 70 mm wide and 2 mm thick. Fatigue tests of SE300 (S3
and S4) and PMR15 (PI and P2) specimens, at room temperature and at 210OC
respectively, at a level of 8000 p.S, located by the peak of the half-sine clamp, have
been carried out. Ultrasonic scans of specimens S3 and S4 before any loading cycles
and after 100, 500, 1000, 2000, 5000, 10000, 20000 and 50000 loading cycles are
shown respectively in Figures l.a-h and Figures 2.a-h. A small delamination,
indicated by lighter areas in the scans, starts to show in both specimens S3 and S4
after applying 500 loading cycles and increases substantially after 5000 loading
cycles. After 5000 loading cycles, the damage area increased more for specimen S4
than specimen S3, which shows that the latter is slightly more fatigue resistant. In
other words, when increasing the temperature from 250C to 210<^C, the resistance to
fatigue slightly decreases. Figures 3.a-h and Figures 4.a-h show the ultrasonic scans
for PMR15 specimens PI and P2 before and after several loading cycles. For both
specimens PI and P2, damage starts after 500 loading cycles and increases
substantially after 20000 loading cycles. At this stage, delamination areas are similar
for both specimens PI and P2 and just a little more pronounced in specimen P2,
which shows that the latter is slightly less fatigue resistant. Hence, an increase in
temperature leads to a decrease in the fatigue resistance properties of the specimens.
From Figures 1, 2, 3 and 4, one can conclude that the PMR15 specimens tested were
slightly more fatigue resistant than the SE300 specimens. In fact, damage in
specimens was generally more defined, clearer and spread more rapidly in the SE300
samples than was the case for the PMR15 samples. Figure 5 shows damage
propagation occuring in an XAS/914 sample (XI), with (0o/±45®/0O)s stacking
sequence, tested at 8000 |iS level and at room temperature, from [3]. Note that
substantial damage existed after 1000 loading cycles in this specimen, which shows
that both SE300 and PMR15 are more fatigue resistant than XAS/914 at room
temperature.
Acoustic fatigue behaviour of panels of the selected materials
Investigations were carried out by installing the CFRP plates in an acoustic
progressive wave tunnel, (APWT) in order to determine the response of CFRP plates
under broadband acoustic excitation simulating jet noise. The plate was fully
clamped around its boundaries on to a vertical steel frame fixed to one side of the
APWT, so that it formed one of the vertical walls of the test section of the APWT.
983
Overall sound pressure levels (OSPL) up to 165 dB of broadband noise in the test
section of the tunnel was generated by a Wyle Laboratories WAS 3000 siren. A
heater panel capable of heating and maintaining the temperature of test plates up to
300*^0 while mounted on the tunnel was designed and built. Temperatures were
monitored and controlled via thermocouples on the panel. Plates were excited by
broadband excitation in the frequency range 80-800 Hz. A B&K type 4136
microphone mounted at the centre of the test section of the tunnel adjacent to the
mid-point of the plate were used for sound pressure measurements. Eight strain
gauges, four on each side of the plate were attached in order to monitor the strain
distribution in the panel while the rig was running; more details of the experiment
may be found in [1]. Acoustic tests were run, at various temperatures and OSPL.
Since plates were excited in the frequency range 80-800 Hz, spectral analyses
would not include the first natural frequency. The natural fundamental frequency of
an SE300 clamped panel was found to be 49.02 Hz analytically. The second -and
third resonance frequencies were 149.5 and 198.5 Hz, as calculated from strain
spectral densities, from tests carried out with an OSPL of 156 dB and a temperature
of 1620C. At 162 dB, results showed that the second resonance frequency was 113
Hz at 150OC and 106.5 Hz at 195^0, which shows that when the temperature and
the OSPL increased, the resonance frequencies of the plates decreased. Also, it was
found that the damping increased at elevated temperatures. The overall modal
viscous damping ratios, for the second mode, were calculated from strain spectral
densities, for an SE300 panel driven at an OSPL of 162 dB and at 1950C, and was
found to be 8.91%; this value is similar to that calculated from analytical
simulations, for the first mode, which is 8,50% at 242^0, (see the analytical section
below). A typical strain spectral density obtained from recorded results is presented
in Figure 6 for an SE300 specimen, at an OSPL = 156 dB and at 1620C, from a
strain gauge in the centre of the specimen. Maximum RMS strain values recorded
from experimental tests, at a strain gauge in the centre of the specimen were, at an
OPSL =156 dB, 1300 \iS, 1800 \iS and 2800 pS at room temperature and at 90^0
and 1620C respectively. These results clearly indicate a trend for significant
increase in dynamic response with increasing temperature.
Experiments were also carried out on a PMR15 panel at various OSPL and
temperatures. Results from tests run at 159 dB and at room temperature show that
the second and third resonance frequencies were indicated as 112 and 182.5 Hz.
When the OSPL increased to 162 dB, the second and third resonance frequencies
decreased to 110.5 Hz and 176.5 Hz. At the same OSPL (162dB) and when
temperature increased to 2810C, the third resonance frequency became 139 Hz. This
984
shows that, for PMR15 plates, when the temperature and the OSPL increased, the
resonance frequencies of the plates decreased. It was also clear that modal damping
increased with increasing temperature. In fact, the overall viscous damping reached
20%, for the second mode, at an OSPL=162 dB and at 2810C. It must be stated here,
however, that apparent damping trends could include nonlinear effects which
influence bandwidths of resonances. Maximum RMS strain values recorded at room
temperature, by a strain gauge in the centre of the plate were found to be: 2700 pS at
153 dB, 2800 pS at 157.9 dB and 2900 pS at 159 dB.
It is clear from these values that increasing the OSPL obviously leads to an increase
of the strain in the plate. Similar results were observed when the temperature was
increased. In fact, at an OSPL of 162 dB, the maximum strain values recorded by a
strain gauge in the centre of the specimen were 3000 pS, 3400 pS and 5000 pS at
105^0, 1650C and 281^0 respectively, which clearly indicates the effects of
temperature. It was observed that both the PMR15 and SE300 panels behaved in a
non linear manner.
Attempts to acoustically fatigue a PMR15 panel were made at 162 dB. No signs of
fatigue damage were shown in an ultrasonic scan of the panel after 1389 minutes of
running time.
3-Analytical work
In order to examine various methods for including damping in a structure, parametric
studies were carried out using the finite element FE method. ANSYS software has
been used. A three-dimensional, 3D layered element, SOLID46 was used to build
theoretical models. The element is defined by eight nodal points, average layer
thickness, layer material direction angles and orthotropic material properties, [4].
Meshes were built in order to carry out modal and harmonic analyses of multilayered
composite plates (410 mm, 280 mm, 2 mm). The plates were fully clamped along all
edges, in order to simulate the panels tested in the APWT. Natural frequencies were
first determined from free vibration analyses and compared to resonance frequency
values derived from experimental data. Then, the plate was driven by harmonic
loading at one point of application. The forcing frequency varied from 0 to 400 Hz.
The amplitude of the load was 50 N. Results for displacements and response phase
angles relative to the force for a chosen position on the plate as a function of
frequency were obtained. The approach was then to carry out parameter studies in
order to examine various methods for including damping in the structure, i.e. to use
highly damped matrix material throughout the whole structure or possible
985
incorporation in a few layers. Structural damping was included, allowing models to
run with different damping values in each ply of the panel. Structural damping is
inherent in the structure and depends on the natural frequency; details on structural
damping modeling may be found in [1]. Analyses were performed considering
structural damping for the first mode. The structural damping was then varied for
plies with the same orientation for a viscous damping ratio ^ = 0.01, 0.02, 0.05, 0.10
and 0.20.
Simulations with SE300
Models were built up with the following stacking sequence ((45°/45°),(0°/90°))s,
lay-up used for the experimental plates. Table 2 gives the first three modal
frequencies of the panel obtained from free vibration analyses results. Harmonic
simulations were carried out and the overall damping value was calculated for each
case with results given in Table 3. As can be seen, if high overall damping is needed
for a structure composed of the SE300 material, increasing the damping value of the
(45°/45®) orientation plies most significantly increases the overall damping value of
the panel. In fact, putting a damping value of 20% in the (45°/45°) orientation plies
leads to an overall viscous damping value of 14.52%, which is better than including
a 10% damping value in all of the plies of the structure.
Harmonic analyses of fully clamped plates were also performed with the values of
material properties taken at several temperatures. Simulations were carried out with
material properties at 2420C and 300OC. Free vibration analyses permitted
calculation of the modal frequencies of the panels at the temperatures mentioned
above. Table 2 also lists the first three modal frequencies from analyses with
material properties at 242^0 and 300^0. The overall viscous damping values,
obtained from FE simulations, are given for each ternperamre in Table 4. Again, the
damping value has been varied through the layers and the overall damping value was
calculated in order to see which of the plies contributes the most to heavily damp the
plate. It was found that putting a damping value of 20% in the (45°/45°) orientation
plies, the first mode viscous damping ratios were 14.62% and 14.55% at 242^0 and
300OC respectively. This shows that this material is more highly damped at high
temperature and presents better damping properties of the two materials at 242^0.
Simulations with PMR15
Free vibration analyses of models built up with the following stacking sequence
((45°/45°),(0°/90®))s were carried out and the first three modal frequencies of the
panel are shown in Table 5. Harmonic analyses were run and the overall damping
986
value was calculated for each simulation with results given in Table 6. If high
overall damping is needed for a structure composed of the PMR15 material,
increasing the damping value of the (45°/45®) orientation plies most significantly
increases the overall damping value of the panel. In fact, putting a damping value
of 20% in the (45°/45°) orientation plies leads to an overall viscous damping value
of 14.39%, while if (0°/90®) orientation plies have a 20% damping value, the
overall damping is 7.42%.
Harmonic analyses of fully clamped plates were also carried out with the values of
material properties taken at several temperatures. Simulations were carried out
using material properties at 3750C and 400oC. Free vibration analyses permitted
calculation of the modal frequencies of the panels at the temperatures mentioned
above. Table 5 lists the first three modal frequencies from analyses with material
properties at 3750C and 400^0. The overall viscous damping values, obtained
from FE simulations, are given for each temperature in Table 7. Again, the
damping value has been varied through the layers and the overall damping value
was calculated in order to see which of the plies contributes the most to heavily
damp the plate. It was found that putting a damping value of 20% in the (45‘^/45°)
orientation plies, the first mode viscous damping ratios were 18.39% and 16.94%
at 3750c and 400OC respectively. This shows that this material is more highly
damped at high temperature and presents better damping properties of the two
materials at 3750C.
4~Conclusions
Two matrix materials, SE300 and PMR15, with potential for use in aircraft
structures in a severe environment, i.e. temperatures up to SOO^C were selected for
this study. Material properties were determined using DMTA techniques and
results show that these materials have high damping abilities at high temperature.
Dynamic loading tests, performed in flexure at room and high temperature showed
that the carbon fibre reinforced PMR15 material is more fatigue resistant than
SE300 and XAS/914 based composites. Acoustic tests using a progressive wave
tunnel, up to a random acoustic OSPL of 162 dB, at room temperature and
elevated temperamres up to 2810C were also performed. When increasing the
excitation level and the temperature higher strain values in the centre of the panels
were recorded. Free vibration and harmonic FE analyses permitted determination
of the natural frequencies and the overall viscous damping values. Resonance
frequencies determined from results obtained from acoustic tests were similar to
987
natural frequencies obtained from FE simulations. Overall viscous damping values
obtained from experimental results agreed well with those obtained from the FE
analyses for SE300 panels. Results obtained for PMR15 panels, from tests, were
higher than those calculated analytically. Both tests and simulations showed that
SE300 and PMR15 present higher damping capabilies at high temperatures.
Conclusions, via parameter studies including material damping, have been drawn
concerning optimal incorporation of a highly damped matrix material into a high
performance structure.
5-AcknowIegments
The authors wish to thank the Minister of Defence for sponsorship of the programme
of research under which the work was carried out. Thanks are also due to Dr M. Nash
of the DRA, Famborough for many helpful discussions throughout the project. _
6-References
1- Benchekchou, B. and White, R.G., Acoustic fatigue and damping technology in
FRP composites, submitted to Composite Structures.
2- Benchekchou, B. and White, R.G., Stresses around fasteners in composite
structures in flexure and effects on fatigue damage initiation: I-Cheese-head bolts.
Composite structures, 33(2), pp. 95-108, November 1995.
3- Drew, R.C. and White, R.G., An experimental investigation into damage
propagation and its effects upon dynamic properties in CFRP composite material .
Proceedings of the Fourth International Conference on Composite Structures, Paisley
College of Technology, July 1987.
4- ANSYS theoretical manual, Swanson Analysis Systems Inc, December 1992.
988
Table La: Loss factor and Young’s modulus values at 40^C and at Tg for
SE300 samples analysed by the DMTA.
stacking sequences
(4.45/-45)4
(-457+45)4
(45/45;0/90)s
(0/90)4
Tg(°C)
242
242
240.3
238.71
T| atTg
0.085
0.085
0.081
0.061
Ti at 40°C
0.012
0.014
0.010
0.0097
Log E’ at40°C
9.870
9.840
9.970
10:097
Table Lb: Loss factor and Young’s modulus values at 40^C and at Tg for
PMR15 samples analysed by the DMTA.
Stacking sequences
(+45/--45)4
(0/90;45/45)s
(45/45;0/90)s
(0/90)4
Tg (OC)
372
375
375
375
Tl atTg
0.117
0.124
0.121
0.129
T| at 40®C
0.0110
0.0138
0.0086
0.0117
Log E’ at 40OC
9.583
9.944
9.972
9.875
989
Table 2: The first three modal frequencies for SE300 panel; analyses carried out
with material propert ies at room temperature, at 242*^0 and at 300°C.
Room temperature
242^0
300OC
49.02Hz
44.72Hz
43.61Hz
155.80Hz
143.27Hz
140.17Hz
212.04Hz
194.19Hz
189.68Hz
Table 3: Overall viscous damping values of SE300 panel. Values are calculated from
results obtained from harmonic analyses; the material damping being considered for
the first mode.
Simulation with
damping of
(45°/45°)orientation
plies
(0®/90°)orientation
plies
5%
3.81%
2.51%
10%
7.06%
3.92%
20%
14.52%
11.57%
Table 4: Overall viscous damping values of SE300 panel, for the first mode.
Temperature
(°C)
Overall viscous damping
25
1.20%
242
8.50%
300
5.45%
990
Table 5: The first three modal frequencies for PMR15 panel; FE analyses carried out
with material properties at room temperature, 375°C and 400°C.
Room temperature
3750c
400OC
43.43Hz
32.68Hz
25.54Hz
130.35Hz
99.65Hz
78.83Hz
183.96Hz
139.04Hz
109.11Hz
Table 6; Overall viscous damping values of PMR15 panel. Values are calculated
from results obtained from harmonic analyses; the material damping being
considered for the first mode..
Simulation with
damping of
(45°/450)orientation
plies
(0o/90O)orientation
plies
5%
3.76%
2.44%
10%
7.26%
4.10%
20%
14.39%
7.42%
Table 7: Overall viscous damping values of PMR15 panel, for the first mode..
Temperature
Overall viscous damping
(OQ
25
1.33%
275
13.24%
400
8.6%
991
a: before any loading cycles b: after 100 loading cycles c: after 500 loading cycles
d: after 1000 loading cycles e: after 2000 loading cycles f: after 5000 loading cycles
g: after 1 0000 loading cycles h: after 20000 loading cycles
Figure 1. Ultrasonic scans of specimen S3 after applying different numbers of loading cycles.
(SE 300 material, ambient temperature)
a: before any loading cycles b: after 100 loading cycles c: after 500 loading cycles
d: after 1000 loading cycles e: after 2000 loading cycles f: after 5000 loading cycles
Fisure 2. Ultrasonic scans of specimen S4 after applying different numbers of loading cycles.
(SE300 material, 210OC)
992
a: before any loading cycles b: after 100 loading cycles c: after 500 loading cycles
d: after 2000 loading cycles e: after 20000 loading cycles f: after 50000 loading cycles
g: after 100000 loading cycles
Figure 3. Ultrasonic scans of specimen PI after applying different numbers of loading cycles
(PMR15 material, ambient temperature, 8000]LlS)
a: before any loading cycles b: after 500 loading cycles c: after 1000 loading cycles
g: after 10000 loading cycles
h: after 20000 loading cycles
Figure 4. Ultrasonic scans of specimen P2 after applying different numbers of loading cycles.
(PMR15 material, 2100C, 8000M.S)
993
a: before any b: after 100 c: after 500 d: after 1000
loading cycles loading cycles loading cycles loading cycles
e: after 2000 f: after 5000 g: after 10000 h: after 20000
loading cycles loading cycles loading cycles loading cycles
Figure 5. Ultrasonic scans of an X AS/9 14 specimen fatigued at a level of 8000 llS
showing the damage propagation; the lay-up is (0/±45/90)s, [3],
0 Lin Hz RCLD 1.6k
Figure 6: SE300 specimen SI strain spectral density, recorded from strain gauge
ST2, OSPL =156 dB, temperature = 162^C.
994
THE BEHAVIOUR OF LIGHT WEIGHT HONEYCOMB SANDWICH
PANELS UNDER ACOUSTIC LOADING
David Millar
Senior Stress Engineer
Short Bros. PLC
Airport Road
Belfast
Northern Ireland
SUMMARY
This paper discusses the results of a progressive wave tube test on a carbon composite
honeycomb sandwich panel. A comparison was made with the test panel failure and the
failure of panels of similar construction used in the intake ducts of jet engine nacelles.
The measured panel response is compared with traditional analytical methods and finite
element techniques.
Nomenclature
= Overall rms stress (psi) or strain (jxs).
7t =3.14159
= Fundamental frequency (Hz).
5 = Critical damping ratio (*0.017).
Lps(fn) = Spectrum level of acoustic pressure (- expressed as a fluctuating rms pressure in psi in a
1 Hz band).
jr = Joint acceptance function (non dimensional).
= Characteristic modal pressure (psi)
ph = Mass per unit area (Ib/in^)
Sic “ Modal stress (psi) or modal strain (|j£).
Wjc = Modal displacement (in),
a = Panel length (in),
b = Panel width (in).
x,y,z = Co-ordinate axes.
1.0 Introduction
Honeycomb sandwich panels have been used for some time in the aircraft industry as
structural members which offer a high bending stiffness relative to their weight. In
particular, they have proved very attractive in the construction of jet engine nacelle
intake ducts where, in addition to their load carrying ability, they have been used for
noise attenuation.
995
2.0 Acoustic Fatigue
The intake duct of a jet engine nacelle can experience a severe acoustic environment
and as such the integrity of the nacelle must be assessed with regard to acoustic fatigue
[1], Acoustic fatigue characterises the behaviour of structures subject to acoustic
loading, in which the fluctuating sound pressure levels can lead to a fatigue failure of
the structure. The traditional approach to acoustic fatigue analysis has assumed
fundamental mode response and given that aircraft panels will in general, have
fundamental frequencies of the order of several hundred hertz, it is clear that the
potential to accumulate several thousand fatigue cycles per flight can exist.
Techniques for analysing the response of structures to acoustic loads were developed
originally by Miles [2] and Powell [3], Other significant contributions are listed in
References 4-7. Design guides such as AGARD [8] and the Engineering Sciences
Data Unit (ESDU) series of data sheets on vibration and acoustic fatigue [9], have
proved useful in the early stages of design.
Note - further details on the general subject of acoustic fatigue can be found in Ref
10, while a more detailed review of the subject up to more recent times is presented in
Ref 11.
3.0 In Service Failures
In recent years a number of failures have been experienced involving intake barrel
honeycomb sandwich panels. Failures have been experienced with panels which had
both aluminium facing and backing skins and carbon composite panels. The metal
intake liner was observed to have skin cracking and also core failure, while the
composite panel was only observed to have core failure.
With regard to the metal panels, flight testing was carried out and the predominant
response frequency was observed to be at the fan blade passing frequency - much
higher than the fundamental frequency of the intake barrel; this went some way to
explaining why the traditional approach in estimating the response did not indicate a
cause for concern. The response of the panel was also very narrow band - almost a
pure sinusoid (again differing from the traditional approach of broad band/random load
and response), and the subsequent analysis of the results was based on a mechanical
fatigue approach [12]. Subsequent fleet inspections revealed that core failure was
observed prior to skin failure and it was assumed that the skin failure was in fact
caused by a breakdown in the sandwich panel construction. The core was replaced
with a higher density variety, with higher shear strength and moduli. This modification
has been in service for several years with no reported failures. The modification
represented only a moderate weight increase of the panel, without recourse to
changing skin thickness, which would have proved very expensive and resulted in a
substantial weight penalty.
As mentioned above, another intake duct, of carbon composite construction, also
began to suffer from core failure. The panels of this duct had a carbon backing skin
while the facing skin had a wire mesh bonded to an open weave carbon sheet. The only
similarity was the use of the same density of honeycomb core (although of different
cell size and depth). For other reasons this core had been replaced by a heavier variety,
prior to the discovery of the core failures and the impact of the failures was minimised.
996
Limited data is available on similar failures and only 2 other cases, regarding nacelle
intake barrels, appear to have been documented [13 & 14], however neither case
involved sandwich panels.
A number of theories had been put forward as to the cause of the failures. These
included neighbouring cells resonating out of phase, cell walls resonating or possibly
the panel vibrating as a 2 degree of freedom system (the facing and backing skins
acting as the masses, with the core as the spring) - this phenomenon had originally
been investigated by Mead [15].
4.0 Physical Testing
A number of tests were carried out with "beam" type high cycle fatigue specimens and
also small segments of intake barrel. None of these tests were able to reproduce the
failures observed in service (Figure 1.0) which further served to reinforce the belief
that the failures were attributed to an acoustic mechanism as opposed to a mechanical
vibration mechanism, however in an attempt to cover all aspects it was decided to
carry out a progressive wave tube (PWT) test on an abbreviated panel.
For simplicity it was decided to test a flat sandwich panel of overall dimensions
36"x21" (Figure 2.0). The panel was instrumented with 12 strain gauges and 2
accelerometers. Two pressure transducers were also mounted in the fixture
surrounding the specimen.
Testing was carried out by the Consultancy Service at the Institute of Sound and
Vibration Research (IS VR) at the University of Southampton.
4.1 PWT Results
The panel was first subject to a sine sweep from 50 to 1000 Hz in order to identify its
resonant frequencies. The response of a strain gauge at the centre of the panel has been
included in Figure 3.0. On completion of the sine sweeps, the linearity tests were
carried out.. As only 8 channels could be accommodated at one time, it had been
decided to arrange the parameters into 5 groups, with each group containing 4 strain
gauges, 2 accelerometers, 1 pressure transducer on the fixture and 1 pressure
transducer in the PWT (this was required by the facility for the feedback loop).
The initial tests were carried out with a power spectral density of the applied loading
constant over the 100 Hz to 500 Hz range, however when using this bandwidth only
155 dB overall, could be achieved. In an attempt to increase the strain levels it was
decided to reduce the bandwidth to 200 Hz. The bandwidth (BW) was subsequently
reduced to 100 Hz and finally 1/3 octave centred on the predominant response
frequency of the panel. When failure occurred a dramatic change in response was
observed. The failure mechanism was that of core failure as shown in Figure 4.0. There
was no indication of facing or backing skin distress.
997
5.0 Comparison With Theoretical Predictions
5.1 Fundamental Frequency
From the strain gauge readings the panel was seen to be vibrating with simply
supported edge conditions. Soovere [7] suggests that "effective" dimensions
(essentially from the start of the pan down) be used to determine the fundamental
frequency which is given by;
x\n
This equation is applicable to simply supported panels with isotropic facing and
backing skins, thus for the purpose of applying the above equation, the actual section
was approximated to a symmetric (isotropic) section. The predicted fundamental
frequency is given below. It was observed however, that if the panel dimensions are
taken relative to mid way between the staggered pitch of the fasteners a significant
improvement was achieved (see "Soovere (2)" in table 1). Alternative frequency
estimations using an FE model and an ESDU data item [16] are summarised in the
following table;
Method
Freq.(Hz)
% Error
Measured
228
-
Soovere
274.3
+20.3
Soovere
(2)
213.3
-6.4
FE
239.04
+4.8
ESDU
193
-15.3
Table 1 - Comparison of calculated frequencies for simply supported sandwich
panel.
Note; the percentage error is based on the actual measured response frequency of the
panel in the PWT.
Given that the excitation bandwidth extended (at least initially) up to 500 Hz, modes
up to 500 Hz were obtained from the FE model. In actual fact 2 FE models were used,
the first was a basic model with 380 elements, however a more detailed model, shown
in Figure 5.0, (with essentially each element split into 4) was used for the results
presented in this paper. The predicted modes from the FE model were as follows;
Mode
No.
Frequency
(Hz)
Mode No. in x
direction (m)
Mode No. in y
direction (n)
Figure
No.
1
239.04
1
1
6
2
334.0
2
1
7
3
430.02
1
2
8
4
1
9
Table 2 - Finite Element Model Predicted Frequencies.
998
5.2 rms Strain
The predicted strains were calculated using Blevins' normal mode method (NMM) [5],
with a joint acceptance of unity for the fundamental mode of vibration, using the
following expression;
s^= Lp.(fJ . ^ (2)
In an attempt to improve the estimated response, the rms strain was calculated for
each mode within the bandwidth of excitation. The Joint accetptance for each mode
was calculated using equation 3 and the calculated strains for each mode were then
factored by the relevant joint acceltance term. The overall strain was then calculated
for all the relevant modes. A comparison with ESDI! [16] has also been included,
however the ESDU method does not provide an indication of shear stress in the core.
Soovere presents a simple expression for the joint acceptance function for a simply
supported panel excited by an (acoustic) progressive wave, for the case where n is
odd;
•2 _ ^ (l-Cos(m7c)Cos(c0ra/c) , .
(1 - (cD^a/ mrcc)^)
Note, when n is even the joint acceptance is zero.
Given that the bandwidth varied for the applied loading, the overall SPLs were
expressed as spectrum levels for the purpose of comparison in the linearity results, the
results (both measured and predicted) have been summarised in table 3, (SGI results
have been plotted in Figure 10.0). The results from the ESDU data item [16] have
been included in table 4 for comparison.
Note - due to recorder channel limitations SGI & SG2 were not connected at the time
of failure and no results were available at the highest sound pressure levels.
OASPL
(dB)
Spectrum
Level SPL
(dB)
Measured
Strains(u£)
SGI SG2
Calculated
(ps) j=l
SGI SG2
Calculated
(Multi Mode)
SGI (us) SG2
130
107
7
7
8.7
9.2
2.5
2.6
140
117
20
19
27.5
29.1
8.0
8.1
150
127
55
60
87.0
92.0
25.4
25.8
155
132
100
100
154.7
163.7
45.1
45.8
157
134
130
130
194.8
206.0
56.8
57.6
163
140
-
-
388.7
411.1
202.5
162.7
164
141
-
-
436.1
461.2
312.0
250.7
Table 3 - Comparison of Measured & Predicted rms Strains
for the Panel Centre, Facing & Backing Skin Gauges.
999
OASPL
(dB)
Spectrum
Level SPL
(dB)
Measured
Strains (pe)
SGI SG2
ESDI! Strains
SGI SG2
130
107
7
7
10.1
29
140
117
20
19
150
127
55
60
101
290
155
132
100
179.5
515.7
157
134
130
■E9
253.6
728.5
163
140
-
-
637
1830
164
141
-
-
1010
2900
Table 4 - Comparison of Measured & ESDI! Predictions of the
rms Strains for the Panel Centre, Facing & Backing Skin Gauges.
There is a considerable difference in the calculated response from using a joint
acceptance of unity for the fundamental mode and that when estimating the joint
acceptance for each mode and calculating the overall response for several modes,
however it was observed that if the average value from both methods is used the
response compares favourably with that measured (-at least for the cases under
consideration). The average value has been included on the linearity plot for SGI,
shown in Figure 10.0). In general, the level of agreement between theory and practice
was considered adequate and it was decided to apply the theory to estimating the shear
stresses in the core (Table 5);
OASPL
(dB)
Spectrum
Level SPL
(dB)
Core Shear
Stress (J=l)
(rms psi)
Core Shear
Stress
(Multi Mode)
(rms psi)
Average
Core Shear
Stress
(rms psi)
Peak Core
Shear
Stress
_ (E£!i _
130
107
0.3
0.09
0.19
0.58
140
117
0.94
0.28
1.84
150
127
2.99
0.88
1.93
5.8
155
5.31
1.57
3.44
10.32
157
134
6.69
1.97
4.33
13.0
163
140
13.35
5.57
9.46
28.38
164
141
14.98
8.59
11.78
35.34
Table 5 - Predicted Core Shear Stress.
6.0 Discussion & Recommendations
The ESDI! method proved very conservative and will thus give a degree of
confidence when used in the early stages of the design process. Blevins Normal Mode
Method was observed to give reasonable accuracy in predicting the highest strains in
the panel and would merit use when designs have been fixed to some degree; at which
stage FE models become available.
For panels whose predominant response is in the fundamental mode it is accepted that
the contribution from shear to overall deformation is very small. The main concern
when designing a honeycomb sandwich panel which is subject to "severe" acoustic
1000
loads has tended to focus on skin strains and to some degree the properties of the core
material have been ignored. The fact that low skin strains are observed has the effect of
giving an impression that there is no cause for concern, however when the properties
of the core material are low or unknown, some caution is required. There is
unfortunately no available S-N data for the type of honeycomb used in the construction
of the panel, however the allowable ultimate strength for the core material is of the
order of 26 psi, so clearly the 163 dB level was sufficient to cause a static failure while
the lower SPLs can be assumed to the have contributed to initiating fatigue damage.
On cutting up the test panel, a large disbond was observed however it did not extend
to the panel edge where cracking had occurred (the mid point of the long edge being
the location of maximum shear for a simply supported panel) and it was the opinion of
the materials department that the failure had not initiated in the disbond.
The SPLs used in the test were not excessively high and were comparable to service
environments (an example of which is given in Table 6). It should be noted that while
the levels in Table 5 are 1/3 octave bandwidths, the actual spectrum is not generally
flat within each band for engine intakes, but is rather made up of tones (Figure 11).
These tones or spectrum levels can thus essentially be the band level and thus some
caution should be exercised when converting intake band levels to spectrum levels
using the traditional approach [17].
1/3 Octave Centre
Sound Pressure
Frequency (Hz)
Level (dB)
100
141
125
133
160
140
200
142
250
140
315
139
Table 6 - Typical Acoustic Service Environment.
Note; Overall levels may reach 160 - 170 dB, however they tend to be influenced by
SPLs at blade passing frequencies, which are much higher than panel fundamental
frequencies.
7.0 Conclusion
It has been shown that although moderate levels of acoustic excitation produce quite
low overall rms strains in the skins of honeycomb sandwich panels, it is still possible,
when using very light weight cores, to generate core shear stresses of a similar order of
magnitude to the allowable ultimate strength of the material.
Acknowledgements
The author acknowledges the support of Short Bros. PLC in the course of preparing
this paper and also the assistance of Mr Neil McWilliam with regard to the FE
modelling.
1001
References
I . 0 Air worthiness Requirements (JAR/FAR) Section 25 .57 1 .d.
2.0 Miles, J.W., "On Stmctural Fatigue Under Random Loading, " Journal of the
Aeronautical Sciences, (1954),Vol.21, p753 - 762.
3.0 Powell, A., "On the Fatigue Failure of Structures due to Vibrations Excited by
Random Pressure Fields,” Journal of the Acoustical Society of America, (1958),
Vol.30, No.l2,pll30- 1135.
4.0 Clarkson, B.L., "Stresses in Skin Panels Subjected to Random Acoustic
Loading," Journal of the Royal Aeronautical Society, (1968), Vol.72,
plOOO- 1010.
5.0 Blevins, R.D., "An Approximate Method for Sonic Fatigue Analysis of
Plates & Shells," Journal of Sound & Vibration, (1989), Vol.129, No.l,
p51-71.
6.0 Holehouse, I., "Sonic Fatigue Design Techniques for Advanced Composite
Aircraft Structures," AFWAL TR 80-3019,(1980).
7.0 Soovere, J., "Random Vibration Analysis of Stiffened Honeycomb Panels with
Beveled Edges," Journal of Aircraft, (1986), Vol.23, No. 6, p537-544.
8.0 Acoustic Fatigue Design Data (Part 1), AGARD-AG- 162-72, (1972).
9.0 ESDU International, London, Series on Vibration & Acoustic Fatigue.
10.0 Richards, E.J., Mead, D.J., "Noise and Acoustic Fatigue in Aeronautics," John
Wiley & Sons, New York, (1968).
II. 0 Clarkson, B.L., "A Review of Sonic Fatigue Technology," NASA CR 4587,
(1995).
12.0 Millar, D., "Analysis of a Honeycomb Sandwich Panel Failure," M.Sc. Thesis,
University of Sheffield, (1995).
13.0 Holehouse, L, "Sonic Fatigue of Aircraft Structures due to Jet Engine Fan
Noise," Journal of Sound & Vibration, (1971), Vol. 17, No. 3, p287-298.
14.0 Soovere, J., "Correlation of Sonic Fatigue Failures in Large Fan Engine Ducts
with Simplified Theory," AGAEUD CPI 13 (Symposium on Acoustic Fatigue),
(1972), pi 1-1 - 11-13.
15.0 Mead, D.J., "Bond Stresses in a Randomly Vibrating Sandwich Plate: Single
Mode Theoiy," Journal of Sound & Vibration, (1964), Vol.l, No. 3,
p258-269.
16.0 ESDU Data Item 86024 (ESDUpac A8624), "Estimation of RMS Strain in
Laminated Face Plates of Simply Supported Sandwich Panels Subjected to
Random Acoustic Loading," Vol. 3 of Vibration & Acoustic Fatigue Series.
17.0 ESDU Data Item 66016, "Bandwidth Correction," Vol. 1 of Vibration &
Acoustic Fatigue Series.
1002
Backing Skin
Figure 2.0 - PWT Test Specimen.
1003
Microstrain (dB)
Figure 3.0 - Response of Strain Gauge SGI During Sine Sweep.
Figure 4.0 - Section Through Failure Region in PWT Panel.
1004
Figure 5.0 - PWT Panel Finite Element Model.
Figure 8.0 - FE Mode 1 (m==l, n=2) Figure 9.0 - FE Mode 2 (m=3, n=l)
1005
Sound Pressure Level (dB)
Figure 10.0 - Linearity Plot for SGI (Measured & NMM Prediction).
Figure 11.0 - Typical Spectral Content of Intake Duct Sound
Pressures with Equivalent 1/3 Octave Levels Superimposed.
1006
Time Domain Dynamic Finite Element Modelling in Acoustic Fatigue Design
Authors:
P. D. Green
Military Aircraft
British Aerospace
Warton
A. Killey
Sowerby Research Centre
British Aerospace
Filton
Summary
Advanced Aircraft are expected to fly in increasingly severe and varied acoustic environments.
Improvements are needed in the methods used to design aircraft against acoustic fatigue. Since fatigue
life depends strongly on the magnitude of the cyclic stress and the mean stress, it is important to be able
to the predict the dynamic stress response of an aircraft to random acoustic loading as accurately as
possible.
The established method of determining fatigue life relies on linear vibration theory and assumes that the
acoustic pressure is fully spatially correlated across the whole structure. The technique becomes
increasingly unsatisfactory when geometric non-linearities start to occur at lugh noise levels and/or
when the structure is significantly curved. Also the excitation is generally not in phase across the whole
structure because of complex aerodynamic effects.
Recent advances in finite element modelling, combined with the general availability of extremely fast
supercomputers, have made it practical to carry out non-linear random vibration response predictions
using time stepping finite element (FE) codes.
Using the time domain Monte Carlo (TDMC) technique it is possible to model multi-modal vibrations
of stiffened aircraft panels without making the simplifying assumptions concerning the linearity of the
response and the characteristics of the noise excitation.
The technique has been developed initially using a simple flat plate model. This paper presents some of
the results obtained during the course of this work. Also described are the results of a study of the
“snap-through” behaviour of the flat plate, using time domain finite element analysis. For simplicity, it
was assumed that the dynamic loading was fully in phase across the plate.
Introduction
Aircraft structures basically consist of thin, generally curved, plates attached to a supporting framework.
During flight these stiffen^ panels are subjected to a combination of static and dynamic aerodynamic
loads. On some aircraft there may be additional quasi-static thermal loads due to the impingement of jet
effluxes in some areas. Parts of advanced short take of and landing (ASTOVL) .aircraft may be required
to withstand noise levels up to 175dB and temperatures up to 200deg C. Under these conditions the
established methods of dynamic stress analysis for acoustic fatigue design are inappropriate and cannot
be employed.
British Aerospace (BAe), Sowerby Research Centre (SRC) and Military Aircraft (MA) have been
developing a method to predict the stress/strain response of aircraft structures in these extreme loading
situations. The primary .consideration has been the requirement to create an acoustic fatigue design tool
for dealing with combined static and dynamic loads, including thermally generated “quasi-static” loads.
1007
The resonant response of thin aircraft structures to aeroacoustic loading is generally in a firequency
range which implies that, if defects form, they will quickly grow. Hence to be conservative, it is
generally assumed that a component has reached its life when it is possible to find quite small defects
by non-destructive evaluation techniques. Several different materials and construction methods are used
in modern aircraft and so there are a number of possible failure criteria. In the case of metals, it is the
presence of cracks larger than a certain size. For composites it can be the occurrence of either cracking
or delamination. Degradation due to the presence of microcracks may be monitored by measuring the
level of stiffness reduction which has taken place.
This philosophy simplifies the type of stress analysis needed, because it is not necessary to model
structures with defects present. Materials can be assumed to have simple elastic properties which
remain unchanged throughout their lives. In consequence, it is necessary to know the fatigue behaviour
in terms of a direct relationship between number of cycles to failure and the magnitude of the “nominal”
cyclic stress, or strain, at a reference location.
If considered important and capable of satisfactory treatment, the relationship can be modified to take
into account material property changes due to the development of very small defects at points of stress
concentration. For example, metal plasticity in the region of a small crack, could be included in an
analysis of the stress distribution around a fastener hole. It is well known that plasticity reduces the
peaks of stress which are predicted at defects by analysis which assumes perfectly elastic material
behaviour.
The technique developed at BAe for modelling high acoustic loads combined with possible thermal
buckling uses the time domain Monte Carlo (TDMC) technique together with finite element analysis by
proprietary FE codes. Response characteristics are predicted directly in the time domain using
simulated random acoustic loadings. These may then be used in fatigue life estimations which employ
cycle counting methods such as Rainflow counting. It is now practical to predict the vibrational
response of stiffened aircraft panels without the necessity to assume a linear response, and without
simplifying the spatial and temporal representation of the noise excitation.
Since the technique uses proprietary finite element codes, quite large and complex models of aircraft
structure can be analysed in a single run. Standard pre- and post-processor techniques are available to
speed up generation of the finite element mesh and to display the stress/strain results.
The initial development work was carried out by modelling the random vibration of a flat plate. For
fully in-phase random loading at low noise levels the predicted response is predominantly single mode
and at fhe frequency calculated by linear theory. However, as the decibel level is increased, the
frequency of the fundamental rises due to geometrically non-linear stiffening. At veiy high dB levels the
predicted response becomes multi-modal; the resonance peaks move to higher and higher frequencies
and broaden.
The effect of static loading on the response has been studied as part of these investigations to assist in
the validation of the methodology being developed.
Thermal Effects
In some flight conditions it is possible for a panel to be buckled due to constrained thermal expansion
and also be subjected to very noise levels at the same time. An example is when a ASTOVL aircraft
hovers close to the ground for an extended period, panels which are initially curved, or thermally
buckled panels may possibly be snapped through from one side to the other by a large increase in
dynamic pressure.
“Snap-through” can be potentially damaging to the structure of an aircraft if it occurs persistently,
because the process is associated with a large change in the cyclic bending stress present at the edge of
a stiffened panel. High performance aircraft must therefore be designed so that snap-through never
occurs in practice.
1008
The dynamic response of curved panels or buckled flat plates is difficult to predict theoretically because
of non-linear effects. The established acoustic fatigue design techniques, which are based on linear
vibration theory, are only able to provide approximate predictions of the loading regimes in which
particular panels might be expected to undergo snap-through.
The TDMC method can be used to model non-linear multi-modal vibrations of stiffened aircraft panels
which are also subject to quasi-static stress. In particular calculations may be carried out in the post-
buckling regime.
With this technique simulated random dynamic pressure loading, with measured or otherwise known
spectral characteristics is applied to a curved, or post-buckled panel and the time domain response
calculated. The magnitude of the dynamic loading may then be increased until persistent snap-through
is observed in the predicted response. This gives the designer the ability to design out the potential
problem by systematically altering the most important parameters in order to identify the critical
regime.
Fatigue Life Estimation
Although acoustic fatigue is a complex phenomenon, it has been established that the life of a component
mainly depends on its stress/strain history. The most important factors in this regard are the magnitude
and frequency of the cyclic strain and the mean level of stress at the likely failure points. On this basis
fatigue life can be estimated by carrying out the three stage operation illustrated in Figure 1.
Stage 1: Determine Loads
A determination of static design loads is relatively straightforward compared to a calculation of th.e full
temporal and spatial dependence of the aeroacoustic pressure on a military jet in flight. This is an
enormous task in computational fluid dynamics (CFD). Designers have to rely on experimental data
which can come from measurements on existing aircraft or from scale model tests of jets, for example.
Existing databases can be extrapolated if the circumstances are similar. Experimental noise data is
usually in the form of power spectral density curves as opposed to time series fluctuating pressures, but
either can be used, depending on the circumstances.
Stage 2: Calculate Stresses
The technique chosen to obtain the stresses clearly depends on how much knowledge there is about the
expected loads. In the early stages of design analytic techniques would be used to establish approximate
sizes and stress levels. However, later on when the design is nearly completion, finite element (FE)
stress analysis can be used to model the effect of random acoustic loading on the parts of the skin which
are likely to be severely affected. These calculations would, of course, be done including the effect of
attached substructure.
The established method of designing against acoustic fatigue uses a frequency domain technique which
relies on the validity of linear vibration theory. The method forms the basis of a number of methodology
documents published by the Engineering Sciences Data Unit (ESDU). Whenever there^^e large out-of-
plane deflections the frequency domain method cannot be used because of the “geometric non¬
linearity”. From a strictly theoretical point of view such analyses have to be carried out in the time
domain, although approximate methods are applied with some success.
The established technique produces inaccurate results for curved panels, buckled panels and for panels
under high amplitude vibration. Geometric non-linearity usually stiffens a structure in bending so there
is a tendency to overestimate the stress levels using the frequency domain technique. This conservatism
is clearly useful from the point of view of safety, but it can lead to possible “over-design”.
Unfortunately this is not always the case when there are compressive static stresses present. The
established method also fails if the phase of the noise varies significantly over the surface of the
structure, which is the case in a number of aeroacoustic problems. The techniques under development
are designed to overcome these problems.
1009
Stage 3: Estimate Fatigue Life
In cases of random acoustic loading it is customary to assume that damage accumulates according to the
linear Miner's rule. Fatigue life is determined from experimental data in the form of stress (or strain
amplitude), S, versus number of cycles to failure, N. If a number of cycles, n, of stress/strain, S, occur at
a level of stress/strain where N(S) cycles would cause failure then the fractional damage done by the
n(S) cycles is n(S)/N(S).
Various methods have been developed for obtaining n(S) from the stress (strain) response. If the
excitation is stationary, ergodic and the response is narrow band random then the function n(S) can be
shown to be in the form of a Rayleigh distribution and the damage sum can be evaluated from plots of
root mean square stress (or strain) against number of cycles to failure. If the statistics of the response
are not Gaussian then it is necessary to count the numbers of stress cycles from the time domain
response and use constant amplitude S/N curves. It is now widely accepted that the best way of
counting the cycles is to use the Rainflow method, [1].
The Loading Regime
The loads on an aircraft may be conveniently divided into static and dynamic.
Loads which vary only slowly are:
a) Steady Aerodynamic Pressure Loading,
b) In-Plane Loads transferred from “external structure”, and
c) Thermal Loads due to Constrained Expansions.
The rapidly varying loads are, of course, the aeroacoustic pressure fluctuations which originate from
any form of unstable gas or air flow.
This division is central to the methodology which has been developed because it enables the modelling
to be carried out in two distinct phases. The, so called, static loads do vary, of course, but the idea is to
separate effects which occur on a time scale of seconds from the more rapidly varying acoustic
phenomena. The aim is to split the loads so that the quasi-static effects can be calculated in an initial
static analysis which does not depend on a particular dynamic loading regime. Any aerodynamic
pressure may be divided into a steady part and a fluctuating part. The natural place to make the cut-off
is at IHz which means that epoch times for TDMC simulations are then of the order of a second. The
epoch time must not be too short because of statistical errors, and it cannot be too long because this
would invalidate the assumption of constant quasi-static loads. In practice, there is another constraint on
the epoch time. The number of finite elements in the model coupled with the premium on cpu time
places an obvious limit on the epoch time.
Comparison of the Time and Frequency Domain Methods
A flow chart comparing the two methods is given as Figure 2. The main difference between the two
techniques lies in the representation of the dynamic loads. The FD method uses rms loadings and
spectral characteristics, whereas the TD method uses the full time series loadings. Gaussian statistics
are, de facto, assumed by the FD method, but this is not necessarily the case with the TD technique.
Application of the frequency domain method requires that the response is dominated either by a single
mode or a small number of modes. To determine whether or not this is the case in practice, a normal
modes analysis must be followed by a determination of the amount of coupling between the excitation
and each mode. This can be determined quite accurately even if there is a certain amount of potential
non-linearity by computing the joint acceptances for each mode, which are overlap integrals of the
mode shape functions with the spatial characteristics of the excitation. Normally these quantities will be
dominated by a few of the low order modes. If there is significant coupling into more than one mode
then it will be necessary to use the TD method instead.
With the time domain technique it is possible to represent the dynamic loads in a way which models the
convection of the noise field across the structure. Very complicated loadings can be applied to large
1010
models but in consequence it can be difficult to validate the results obtained, because they cannot be
checked against anything other than test data which is itself subject to confidence levels. In addition it
must be remembered that the TDMC results themselves are subject to statistical variability. Finally it
should be noted that TDMC data must be used in conjunction with constant amplitude endurance data.
Rms fatigue data can only be used with frequency domain results.
Time Domain Finite Element Modelling
Until recently, the majority of finite element analyses were applied to static loading conditions or “low
frequency normal modes analysis”. The method involves the use of an implicit code to invert in one
operation, a single stiffness matrix, which can be very large. The general availability of extremely fast
super-computers has now made it possible to carry out large scale non-linear dynamic finite element
modelling using explicit FE codes. These codes use very similar types of element formulation to the
implicit ones, e.g. shells, solids and bars, but the solution is advanced in time using a central difference
scheme.
One potentially very useful capability of time domain modelling is the application of acoustic pressure
loadings which vary both in the time and spatial domains. If the spectral characteristics are known,
either from test or from other modelling it is possible to generate samples of random acoustic noise and
apply these directly to the finite element model as a series of “load curves”.
The technique for determining time series noise was developed by Rice [2] and Shinozuka [3]. They
showed that homogeneous Gaussian random noise can be generated from the power spectral density as
a sum of cosine functions with different frequencies and random phase. Noise can be temporally and
spatially correlated noise by deriving phase differences from cross spectral functions if they are known.
The TDMC method can be quite costly in terms of central processor unit (cpu) time because the
solution must be recalculated at each point in time. To reduce execution times, the explicit codes
employ reduced numbers of volume integration points in the finite element formulations. However in
this work cpu times are extended because long epoch times are required to ensure adequate statistics. It
can take more than 24hrs to obtain a solution over a half second epoch if there are a few thousand
elements in the model.
Hence there is always a practical limit to the size of a particular time domain finite element analysis,
(TDFEA). If the loading and geometry are not too complicated, the frequency domain method of
analysis can be tried initially to gain more understanding of the nature of the response in an
approximate way. In some cases the vibrational response regime must be considered carefully to decide
whether TDFEA is really necessary. These may be situations where the non-linear effects are only
moderate.
It would be ideal if the full dynamic response of an aircraft could be determined with a fine mesh model
in one huge operation, but experience has shown that this requires too many elements. It is possible to
construct frill models with reduced stiffness using superelements, enabling flutter and buffet to be
studied, because these are essentially low frequency phenomena. However, in time domain analysis it
has been found that models containing a large amount of detail, such as fasteners and individual
composite material plys, require a great deal of cpu time. To progress we must devise some strategies to
overcome this situation. Since a full TDFEA can only handle a part of the aircraft structure, it is very
important that loads external to the area under consideration are properly taken into account. This is
cmcial to the success of this type of modelling as it is to all finite element modelling.
The most important parameter in any time series analysis is the time step. This is determined by the
velocity of sound waves in the structural material, and is generally of the order of a/v where a is the
shortest element dimension and v is the velocity of longitudinal sound. A small time step is therefore
required when the elements are small and the velocity of sound is large. For an aluminium model with
10mm square elements the time step is about 1.6ixs. Hence a TDMC run with a half second epoch time
needs about a half a million steps. A simple 5000 shell element calculation on a Cray C94 would take
approximately 10 hours.
1011
Dynamic FE models of aircraft structure can be constructed in many ways, using shell elements, beam
elements and/or solid elements. Special elements exist for damping and for sliding interfaces. Joints can
be modelled with sliding interfaces, or with short beams, or just with tied nodes. Fasteners can be
modelled with small solid elements, with short beams or with tied nodes, also. Unfortunately, however
short beams and small solid elements cause a dramatic lowering of thC time step. For example, if the
smallest fastener dimension is, say 3mm, the time step will have to be reduced to about O.Sfis if solid
elements are used in the model. The effect on cpu time is such as to make the calculations impractical.
Sliding interfaces are an efficient way to model skin/substructure contact in explicit analyses, but it is
important to choose the algorithm carefully because some techniques can consume large amounts of cpu
time.
The best practical way of representing stiffened aircraft panels for TDMC analyses is considered to be
with four noded shell elements simply tied together at their edges. A number of efficient shell
formulations are avail-able and meshes can be rapidly produced from the design geometry. Of course,
such models cannot be expected to produce highly accurate stress data in the region of small features
but this aspect has to be sacrificed in the interests of achieving statistically significant amounts of time
series data. To improve the accuracy of stress predictions in the neighbourhood of stiffeners etc., it will
be necessary to couple TDMC analyses with fine mesh static analyses.
The Generation of Time Series Data
A number of factors must be borne in mind when generating time series data for TDMC calculations. It
is important to consider carefully the frequency range and number of points which define the load
spectrum in conjunction with the epoch time and number of points on the time series.
The Nyquist Criterion [4] states that the time increment must be less than or equal to one over twice the
upper frequency on the power spectral density curve. For the sake of argument, take the upper
frequency to be IkHz. This means that the time increment must be less than 500ps. A more
conservative time increment is based on the requirement to represent the dynamic response of the
structure as accurately as possible over a full cycle. Assuming a resonant frequency of 500Hz, which is
perhaps near the limit in practice, and 10 points per cycle which is more than sufficient, the lower limit
on the time step works out at about 2C)0|is. Taking all these factors into consideration, the number of
points on the spectrum curve should be of the order of 1000 and there should be between 1000 and
5000 on the time series. Longer epoch times can be used but for reasons of practicality and statistics it
is better to run more than one short epoch simulation rather than one long simulation.
Explicit FE modelling frequently requires that the time step be smaller than 200jj,s. In the example
given above the time step required by the explicit code was 1.6fis. Under these circumstances the
random noise could be defined with a smaller time increment, but going to this level of effort has been
found to produce no measurable change to the calculated response.
Static Initialisation
There are two possible ways of dealing with the effect of static loads in TDMC modelling. Firstly the
complete analysis can be carried out using the explicit code. To do this it is necessary to apply only the
static loads to the model and run the code until equilibrium is reached. By introducing a high level of
artificial damping the stresses created can be relaxed in a relatively short period of time. The time
required depends on the lowest resonant frequency of the structure and the size of the smallest element
in the model. This facility is termed “dynamic relaxation”.
The alternative is to make use of another facility in the explicit code called “static initialisation”. The
deformed shape and stress state of the structure with just the static loads applied are first obtained very
quickly using an implicit code. The solution for the stressed state is then initialised into the explicit
code prior to the application of the dynamic loads. Dynamic relaxation may be used to smooth out any
differences between the models.
1012
Damping Representation
Vibrating aircraft structures are damped by several mechanisms, for example friction at joints, re¬
radiation of acoustic waves, and energy loss in viscoelastic materials It is difficult to generalise about
the relative importance of each damping process in practice. Also reliable quantitative data is not
available in sufficient detail to justify the inclusion of complex models of damping into the TDMC
analyses. Test results on vibrating stiffened aluminium panels tend to show that the damping is best
approximated by a combination of mass and stiffness proportional coefficients. There is a range of
frequencies in which the damping ratio can be considered to be roughly constant. Until more detailed
experimental data are available the most expedient approach is to assume a nominal value for the global
damping ratio which does not change with frequency. Over the years it has become standard practice to
assume a damping ratio of about 2% for fastened aluminium structures.
Equivalent Linearisation
There are some loading regimes in which the non-linear response to high levels of random acoustic
loading can be approximately found using a linearisation technique combined with a frequency domain
analysis. The basic idea is to replace the non-linear stiffness term in the general vibration equation by a
linear term such that the difference between the rms response of the two equations is minimised with
respect to a shifted fundamental resonance frequency. If an approximate equation for the non-linear
stiffness is known then it is possible to derive an expression for the shifted “non-linear” resonance
frequency. The rms response to random acoustic loading may then be found by combining the
Miles/Clarkson equation with some form of static geometrically non-linear analysis. References to this
technique are Blevins [5], Mei [6] and Roberts & Spanos [7]. Where the geometry is complex the most
appropriate form of analysis is clearly finite element analysis.
Implementation and Validation Studies
The stress analysis work described in this paper has been undertaken using MSC-NASTRAN and
LLNL-DYNA.
NASTRAN is a well known implicit finite element code which is capable of handling very large
numbers of elements. It has been developed very much with aerospace structural analysis in mind. It is
basically a linear analysis code, although there are a large number of adaptations to deal with non-linear
problems. It can also function as a dynamic code, but is much slower than DYNA in this mode because
it basically needs to solve the complete problem at each time step. The non-linear features which are
most relevant to the type of stress analysis being discussed here are those concerned with geometric
non-linearity. Geometric non-linearity is treated by dividing the load into a series of steps, obtaining the
solution incrementally. In this work, the code has been used for linear and non-linear static analyses and
for normal modes analyses.
DYNA is an explicit finite element code originally developed for the calculation of the non-linear
transient response of three dimensional structures. The code has shell, beam and solid element models
and there are a large number of non-linear and/or anisotropic material models available. DYTMA was
developed primarily for the modelling of impact and there is no limit, as far as the code is concerned,
on the size of finite element model which can be analysed. Many of its advanced features relate to
impact modelling and are not required for this work. One useful feature, however is the laminated
composite material model based on the equivalent single layer approximation. This code has been used
for the TDMC calculations presented in this paper.
Finite element models for the stress analyses were produced using MSC-PATRAN, It has a wide range
of geometry and mesh generation tools and now has built in interfaces for both NASTRAN and DYNA.
The element definitions are compatible with both codes and it is a simple matter to toggle between the
two codes by changing the analysis preference. Not all the features of DYNA are supported and some
of the parameters must be set by editing the DYNA bulk data produced by PATRAN.
NASTRAN results were post-processed using PATRAN. DYNA results were post processed using
TAURUS, which is faster and easier to use than PATRAN for this task; Some special in-house codes
have been written to generate random acoustic noise from power spectral densities, as described above,
1013
and to post process time series output from TAURUS. One of the codes incorporates a fast Fourier
transform (FTT) routine to determine spectral responses from the DYNA time series predictions. These
codes are covered under the generic title “NEW-DYNAMIC”.
TDMC Calculations on a Simple Flat Plate
Calculations have been performed on a very simple model to implement the TDMC technique and
develop the in-house software referred to above. A PATRAN database was constructed representing a
simple flat plate, 350mm x 280mm x 1.2mm thick as an array of shell elements 34x28. For simplicity
the boundary conditions were taken either as simply supported or clamped. There are a number of
alternative shell element formulations available in DYNA, [8]. The Hughes-Liu shell was used initially
because of its good reputation for accuracy, but later a switch was made to a similar, but slightly faster
shell element, called the YASE. It was found that equally satisfactory results could be obtained more
quickly using this element.
Analyses without Static Loads
A series of DYNA calculations were carried out with a fiiUy correlated random acoustic pressure load
with a flat noise spectrum between OHz and 1024Hz. Investigations were carried out into the effect of
varying the sound pressure level, the epoch time, the mesh resolution, the damping coefficient and the
stochastic function.
Figure 3 shows the displacement response of the central node of the model for a sound pressure level of
115dB (about 12Pa rms), simply supported edges and mass proportional damping set so that the
damping ratio was equal to 2% at the fundamental (1,1) resonance of the plate. The corresponding
spectral response is shown in Figure 4. shows a sharp resonance peak at a frequency of 61.0Hz which is
very close to the theoretical frequency of the (1,1) mode for the simply supported plate. The in-phase
loading means that only the modes with odd numbered indices are excited. I^e peaks corresponding to
the (3,1) and (1,3) modes are, however, not visible on the plot because they are too small. It may be
concluded from these results that the behaviour of the plate at these pressure level is well within the
linear regime.
An investigation into the behaviour of the rms displacement response as a fijnction of SPL was carried
out by increasing the loading incrementally from 75dB (0.1 2Pa rms) to 175dB (12kPa rms). The results
are shown in Figure 5. Also shown are theoretical predictions obtained using the Miles/Clarkson
formula with NASTRAN linear and non-linear analyses as explained above, see below for discussion.
The statistical variation of the results was investigated by repeating a half second epoch TDMC run ten
times with different samples of flat spectrum noise. It was found that the standard error of the rms
response was about 16%. A second set of ten repeats were carried out with the epoch increased to 2.5s.
In this case the standard error reduced to roughly 8%. From the theory of stochastic processes, it can be
shown that the standard error is inversely proportional to the square root of the epoch time. On this
basis therefore the ratio between the standard errors should be equal to the square root of five, or 2.23.
From the analyses this ratio is about 2. Further runs established that these results are not affected by the
vibration amplitude, even when the response becomes non-linear.
Cautiously therefore, it can be concluded that the variance of the TDMC results is unaffected by non¬
linearity of the response. This is an important finding because it builds confidence in the technique. In
many practical situations it may be necessary to rely on just one simulation and an appropriate factor of
safety. It can be quite time consuming to carry out a large number of repeat TDMC simulations. The
level of variance would be first established by repeating one load case a number of times, before
confidently applying it to the results of other load cases.
Comparisons with Linearised Theory
The linear theory of plate bending, [9], leads to relationships between the central deflection, w, of a
rectangular plate and a uniform static pressure load, Pstat which take the following form.
Psutab = kcffW (1)
1014
where a and b are the length and breadth of the plate, and k^ff is an effective stiffness parameter which is
a function of the modulus of rigidity of the plate and the edge boundary conditions. For the plate
studied k^ff is about 30N/mm for the case of simply supported edges and lOON/mm for clamped edges.
The above equation only holds, however, at very low amplitudes, as can be seen from Figure 6. This
compares geometrically non-linear NASTRAN predictions with the linear ^eory. Curves are shown for
both simply supported and clamped boundary conditions. The finite element results show the
characteristic hardening spring type of non-linearity.
At higher amplitudes the dynamic behaviour may be approximately predicted using “equivalent
linearisation” theory, which assumes that the response remains predominantly single mode, but with a
resonant frequency which rises as the stiffness of the structure increases. When the deflection is large
the static force-deflection relationship can be written as the sum of a linear stiffness term and a cubic
non-linear term:
psiatab = kw(l+pw^) (2)
where a b is the force, k is the linear stiffness. The equation is written with the leading term factored
out to emphasis the point that p. is a constant which is small compared to the rms deflection. In the limit
of small w we can expect the pw^ term in the brackets to be negligible compared to one, which means
that the k in this equation must be the same as kcff above.
Equation 2 was fitted to the NASTRAN results shown in Figure 6 to find the best fit values of k and p.
Table 1 shows the results compared with the effective stiffness calculated from linear plate bending
theory. It can be seen that the theoretical stiffness is almost identical to the best fit k from the non-linear
finite element analysis.
With reference to the results in Figure 5, it is obvious that the nature of the response is strongly
dependent on the amplitude of the vibrations. For rms displacements up to about 4% of the plate
thickness the behaviour was completely linear. For displacements between 4% and 150% of plate
thickness, the response was essentially single mode dominated but the level could not be predicted by
the Miles/Clarkson approach. The “equivalently linear” solution does, however, agree with the DYNA
result up to a displacement of about l.8mm. The linearisation approach cannot be expected to be
correct for displacements above about 1.5 times plate thickness. Above this point the response predicted
by DYNA was multi-modal and strongly non-linear. The equivalently linear predictions departed
considerably from the DYNA results when the vibration amplitude was very high.
It was also observed that the frequency response peaks became increasingly noisy for higher pressures,
representing the increased level of non-linearity in the plate vibrations. The increase in the frequency of
the fundamental mode with acoustic pressure, as calculated by DYNA, is shown in Figure 7. Predictions
from equivalent linearisation theory and from the theory of Duffing's equation are also included, see
Nayfeh & Mook [10].
t„, = f(l + 3nw™,Y’
Equiv. Lin. Pred.
(3)
f„, = f(l+(3/8)w„,^)
Duffing's Eq. Pred.
(4)
The DYNA results lie mostly between the two theoretical curves, agreeing particularly well with the
results of equivalent linearisation theory up to around 700Pa (151dB rms). The level of agreernent
obtained shows that the frequency response behaviour of the DYNA model is similar the theoretical
predictions, providing an independent check on the results. As might be expected, at around 700Pa the
agreement begins to breakdown, since the linearisation theories are not valid for deflections which are
significantly greater than the plate thickness. It may be concluded, however, that the effect of geometric
non-linearity at high amplitudes is being computed by DYNA in a reasonably accurate manner. A
detailed comparison with experimental data is needed to determine the accuracy of the DYNA response
predictions themselves.
1015
Due to the increasingly irregular shape of the frequency response functions derived from the DYNA
time series predictions at high acoustic loads, it was not possible to calculate very accurate peak widths
for pressures above approximately 135dB (I20Pa rmsX Figure 8 shows that the width of the peak
increased with increasing acoustic pressure, but not in a regular manner. When the damping is mass
proportional, equivalently linear theory predicts that the width of the peak should remain unchanged as
the pressure rises. This is because the geometric stiffening effect of rising acoustic pressure exactly
cancels the effect of a smaller damping ratio at the higher resonant frequency. This graph shows this as
a horizontal straight line at 2.44Hz. The DYNA result is closer to the type of behaviour observed
experimentally where the width of the peak generally increases with increasing the sound pressure level.
Analyses with Combined Loads
Further work was conducted with static loads superimposed on different levels of random acoustic
loading. These calculations were done using the coupled NASTRAN-DYNA approach outlined earlier.
That is to say the deformed geometry was obtained by applying the static loads to a NASTRAN model,
with the results being initialised into DYNA and dynamically relaxed before the dynamic loading was
applied. Calculations were performed with compressive in-plane loading, static pressure loading and
thermal loading. With the exception of the thermal runs, the boundary conditions used in these runs
were identical to clamped, except that symmetrical in-plane movement of the edges was permitted. We
have called these conditions “semi-clamped”. It has been found that the fundamental resonant frequency
of the plate without static loading is only reduced by a very small amount if the appropriate in-plane
degrees of freedom are released, see Figure 9. These boundary conditions are actually closer to those
which exist in reality when a panel in built into a larger structure.
Figures 10-14 show results of some of the analyses which have been carried out. They give time series
data along with spectra responses calculated by the in-house post-processing code. Numerical data
derived from these results are summarised in Tables 3-5.
A series of analyses have been carried out with compressive in-plane loads equal to one third of the
theoretical buckling loads in compression. For the plate used, the forces per unit side length were -
3.46N/mm in the x-direction and -5.46N/mm in the y-direction. The results of one analysis are shown in
Figure 10. It has been found that the response remains dominated by the fundamental (1,1) mode as
long as the plate is unbuckled and the SPL is low. The softening effect of the compressive loads on the
frequency agreed quite well with Rayleigh-Ritz predictions, [5], up to an SPL of ?dB. At higher sound
pressure levels, the DYNA results reflected stiffness changes which were greater than those predicted
by the theory. The- same was found in the case of tensile loading. It is believed that these differences are
due to approximations built into both the Rayleigh-Ritz theory and the DYNA code.
Figure 1 1 shows the results of a calculation with a superimposed normal pressure. The magnitude of the
pressure, 700Pa, was chosen so as to provide an example of “post-buckled” analysis. This size of
pressure causes the plate to bow out in the centre by about 0.6mm. It is well known that in the post-
buckling regime the random response of a plate depends upon the magnitudes of both the static and
dynamic loads. In this case the static loading was large compared to the applied dynamic loads and
“snap-through” did not occur. The plate simply vibrated about its statically deflected position in the
fundamental mode with a slightly increased frequency.
To provide a test of the DYNA thermal stressing capability, and to carry out an investigation into
“snap-through”, several analyses were carried out with a uniform temperature rise of lOdeg C applied to
the plate with clamped edges. This is quite sufficient to cause buckling because the resulting
compressive biaxial stress, c, is well above the buckling level, Gb- If f is the frequency of the
fundamental and J is a constant equal to 1.248 because of the clamped boundary condition, the two
stresses can be determined approximately from
a = EaT/(l-v^)
(5)
Cb = 4pa^f2/J
(6)
1016
where E, a, v and p are Young's modulus, coefficient of thermal expansion, Poisson's ratio, and density
respectively. Using these formulae we find c = 24MPa and Ob = 14MPa.
Analyses were carried out with several different levels of dynamic load. The results of three of the
calculations are shown in Figures 12,13 and 14. It was found that the threshold for snap-through
occurred at an acoustic load of about IkPa, see Table 4. Below this level the mean deflection, w, is a
function of the static load alone, equal to about 2.8nun (the negative sign indicates that the plate has
bowed in direction of negative z). At higher SPLs the mean deflection reduces because the plate snaps
backwards and forwards between positive and negative z. The calculated response spectra for these
higher level runs, show an additional peak at a very low frequency, ie less than lOHz. This is an artifact
caused by the snap-through since the fundamental resonance of the clamped plate is at 1 13Hz.
Figures 10-14 all show probability density functions derived from the time series data. The fluctuations
on these plots are caused by the smallness of the epoch time. In all cases, except for the thermal
calculations with the two largest acoustic loads, it can be seen that the functions are basically Gaussian
in shape. It may therefore be concluded that it is reasonable to assume that the response of a plate in the
post-buckled region is Gaussian unless there is a large amount of snap-through.
Discussion
The work described is the starting point for investigations and validations using more complex FE
models. Further work has been carried out using models including curvature, sub-structure and detailed
features. It is difficult to validate the predictions obtained from such models by comparing with test data
because the results themselves are open to interpretation. It has been found that the predicted stress
levels are closer to the test results when the chosen location is away from any small features. The lack
of good agreement in the neighbourhood of the features can be explained by the relatively coarse mesh
used in the dynamic models. The overall level agreement was much better than that between predictions
based on linear or equivalently linear theory and test. On the basis of experience, the latter tend to over¬
predict by upwards of factors of two and three. From this work it has been found that the DYNA
predictions tend to be greater than test by amounts which vary but are generally much less. The average
over-prediction was about 40% with a significant change as a function of location.
TDMC runs can take a significant amount of computer time to carry out and it is believed that to make
further improvements the technique should be combined with detailed stressing using static finite
element analysis. Inaccurate results can be obtained if the boundaries of the part of the structure under
analysis are not properly restrained. In the case of models of aircraft panels this may significantly affect
the resonant frequencies which in turn affects the level of calculated dynamic stress. In-plane loads on a
panel, perhaps due to thermal stressing, can alter the fundamental by as much as 100-200Hz. Looked at
from a theoretical point of view, the only way to solve this problem is to construct a second, coarse
model of the component, along with some of its surrounding structure. An initial calculation can then be
carried out with this model in order to obtain the loads and boundary conditions for subsequent
application to the original model.
The dynamic phenomenon of “snap-through” cannot be modelled using existing methods and so the
TDMC / finite element technique offers the engineer a way to determine where the likely regions of
unstable vibration are located in circumstances where the structure is complicated by attachments etc.
Conclusions
This paper has sought to explain how time domain finite element modelling can be used to assist in the
design of aircraft against acoustic fatigue. Although the technique is computationally intensive, it does
have a place in the effort to understand complex vibrations, such as the response of structures to
spatially correlated jet noise excitations, or interactions between high sound pressure levels and thermal
loads.
The work at BAe is continuing in an attempt to provide the analyst with a greater ability to determine
dynamic stress levels in advanced structures with complex loadings.
1017
References
1. Dowling N. E. , Fatigue Prediction for Complicated Stress Strain Histories, J Materials 1, 71
(1972).
2. Rice, In Selected Papers on Noise and Stochastic Processes, Ed N Wax pplSO, Dover New York
(1954).
3. Shinozuka M. , Computers and Structures, 2, 855, (1972).
4. Bendat J. S. and Piersol A.G. , Engineering Applications of Correlation and Spectral Analysis
Wiley (1990).
5. Blevins R.D, , An approximate method for sonic fatigue analysis of plates and shells, J Sound and
Vibration, 129, 1, 51 (1989).
6. Mei C. and Paul D.B. , Non Linear multi-modal response of a clamped rectangular plates to
Acoustic Loading, AIAA Journal, 24, 634, (1986).
7. Roberts J. B. & Spanos P.D , Random Vibration and Statistical Linearisation, Wiley, (1990).
8. Whirley R.G. and Engelmann B.E. , DYNA3D: A Nonlinear, Explicit, Three Dimensional Finite
Element Code for Solid and Structural Mechanics — User Manual, Lawrence Livermore National
Laboratory, UCRL-MA- 107254 Rev. 1, (Nov 1993).
9. Szilard R. , Theory and Analysis of Plates, Prentice Hall, New Jersey.
10. Nayfeh A.H. & Mook D.T. , Non Linear Oscillations, Wiley (1979).
1018
Parameter
lUQgnnggll
HISSH
Linear Theory
NASTRAN
NASTRAN
Simply Supported
30.0
30.7
1.09
Clamped
104.
101.
0.266
Table 1: Values of Parameters fitted to NASTRAN results compared with the
linear theory values.
Static Loads
Acoustic Load
Rrms
(Pa/dB)
Theory ]
1 DYNA
(N.mm
N,
(N.mm
Freq
(Hu)
Wfim
(mm)
■SI
Wrms
(mm)
None
1.2 (95.6)
jnggi
0.00290
114
0.01.2
-3.46
1.2 (95.6)
0.00316
94.1
0.00313
-3.46
643.5 (150.1)
1.69
115
1.08
-3.46
1.2 (95.6)
68.33
0.00359
68.1
0.00368
-3.46
700 (150.9)
68.33
2.09
103
1.42
Table 2: Summary of results of calculations with random acoustic loading
superimposed on compressive in-plane loads.
Static Load
p(Pa)
Acoustic Load
(Pa/dB)
DYNA
None
1.2 (95.6)
114
0.0000
0.01.2
700
None
—
-0.646
—
700
12 (115.6)
115
-0.655
0.0189
700
700 (150.9)
134
-0.477
1.28
5 k
None
—
-2.89
—
5 k
12 (115.6)
179
-2.93
0.0152
5 k
700 (150.9)
173
-2.70
0.999
Table 3: Summary of results of calculations with random acoustic loading
superimposed on static pressure loads.
Temperature
T (<»C)
Acoustic Load
(Pa / dB)
1 DYNA 1
None
1.2 (95.6)
114
0.00256
10
None
—
-2.85
—
10
1.2 (95.6)
234
-2.86
0.0140
10
700 (150.9)
219
-2.80
0.197
10
Ik (154.0)
9.01
2.47
1.02
10
1.2k (155.6)
5.01
-0.717
2.24
10
1.5k (157.5)
3.00
-0,376
2.20
10
2k (160,0)
92.2
0.0185
1.95
10
4k (166.0)
195
0.0568
2.08
Table 4: Summary of results of calculations with random acoustic loading
superimposed on a thermal load.
1019
Figure 2: Flowchart Illustrating the Frequency and Time Domain Techniques
(uiUi),ueujeoB,ds,a ^ O
Displacement of central node for 12Pa rms acoustic pressure,
1.
DYNA
Figure 4: Spectral Response of the fiat plate corresponding to Fig 3. Central
Node for 12Pa rms pressure, DYNA calculation.
3
2.5
2
1
o
^ 1
0.5
0
Figure 5: Rms Central Deflection of the plate versus sound pressure level
Comparison between DYNA results and linear theory.
3
2.5
2
■£
£
E
I
I
o
1
0.5
0
Figure 6: Central deflection of the plate versus pressure, NASTRAN calcula¬
tions compared to linear theory.
500 1000 1500 2000 2500 3000 3500
Static Pressure (Pa)
rms Pressure (Pa)
1022
Figure 7: Variation of fundamental frequency of the (1,1) mode with rms
pressure, Comparison between DYNA and theory.
Damping 2%; Stoctiastic Fn #1 ; 3Sx29 Nodes
0 - 1 - 1 - 1 - 1 - -
0 200 400 600 800 1000 1200
rms Pressure (Pa)
130 140 145 150 153 155
SPL (dB)
Figure 8: Variation of width of the (1,1) mode resonance peak with rms pres¬
sure, Comparison between DYNA and theory.
1023
Figure 9: DYNA model predictions for random vibration of the plate with
semi-clamped boundary conditions.
Figure 10: Random Vibration results with compressive load in the y-direction
of -3.46N/mm and SPL of l.OPa.
Figure 11: Random Acoustic Loading of SPL=12Pa superimposed on a static
pressure of TOOPa.
1024
Figure 12: Random acoustic loading of SPL=700Pa superimposed on a thermal
load of 10 deg, clamped edges. — No Snap Through
Figure 13: Random acoustic loading of SPL— 2kPa superimposed on a thermal
load of 10 deg, clamped edges. — Nearly continuous snap-through.
Figure 14: Random acoustic loading of SPL=4kPa superimposed on a thermal
load of 10 deg, clamped edges. — Dominant acoustic load.
1026
SYSTEM IDENTIFICATION II
ROBUST SUBSYSTEM ESTIMATION
USING ARMA-MODELLING
IN THE FREQUENCY DOMAIN
by U. Prells, A. W. Lees, M. 1. Friswell and M. G. Smart,
Department of Mechanical Engineering of the University of Wales
Swansea,
Singleton Park, Swansea SA2 8PP, United Kingdom
ABSTRACT
This paper reflects early results of the research on modelling the influence of
the foundation on the dynamics of the rotor. The foundation is connected to
the rotor via journal bearings. Dynamic models exist for the subsystems of
the rotor and of the bearings; the first is reliable but the latter is uncertain.
The foundation model is unknown and has to be estimated using rundown
data.These are measured responses of the foundation at the bearings due
to unbalance forces of the rotor which are assumed to be known. Uncer¬
tainties in the bearing model will be transfered to the estimated foundation
parameters. The main scope of this paper is to introduce a method which
enables the decoupling of the problem of model estimation and the problem
of the influence of the bearing model uncertainty.
The influence of changes in the model of the bearings on the estimation
of the foundation model is mainly due to the sensitivity of the computed
forces applied to the foundation at the bearings. These are used together
with the associated measured responses to estimate the foundation model in
the frequency domain. Using an ARMA model in the frequency domain it is
possible to estimate a filtered foundation model rather than the foundation
model itself. The filter is defined in such a way that the resulting force has
minimum sensitivity with respect to deviations in the model of the bearings.
This leads to a robust estimation of the filtered model of the foundation.
Since the filter can be defined in terms of the models of the rotor and of the
bearings only, the problems of estimating the foundation’s influence and of
the sensitivity of the estimates with respect to the model of the bearings
are decoupled.
The method is demonstrated by a simple example of a single-shaft rotor.
Even if the errors in the bearing model are about 50 % the relative input and
output errors of the filtered foundation model are of the same magnitude
as the round-off and truncating errors.
1 INTRODUCTION
An important part of a machine monitoring system for fault diagnostics of
1027
a turbo generator is a reliable mathematical model. This model includes
the subsystems of the rotor, the bearings and the foundation. The model of
the rotor represents the most reliable knowledge, the model of the journal
bearings is uncertain, and despite of intensive research it is not yet possible
to define a model for the foundation which refiects the dynamical contribu¬
tion to the rotor with sufficient accuracy. The first step to determine the
contribution of the foundation on the rotor’s dynamic performance is the
estimation of a reliable foundation model.
Rundown data are available, i.e. displacements ufb{<->^) ^ of the
foundation at the bearings which are due to an unbalance force /c/(a;) G
of the m-shaft rotor, given at discrete frequencies a; € := {wi, ■ • • , wm}?
and this data may be used to estimate the foundation model. A com¬
mon method ([1],[2],[3],[4],[5]) is to estimate the unknown dynamic stiffness
matrix 6 i^4mx4m foundation at the bearings using the in¬
put/output equation
F{uj)upb{(^) = (1)
where the force fpB of the foundation at the bearings can be expressed
by dynamic condensation in terms of the data ups, fu and in terms of the
dynamic stiffness matrices Ar,B of the rotor and the bearings respectively
yielding
fpB = -Bufb + [0, B]A~^q ^ ^ . (2)
Here the dynamic stiffness matrix Arb of the rotor mounted on the bearings
is partitioned with respect to the n inner degrees-of-freedom (dof) of the
rotor and to the 4m connecting (interface) dof
^RB
Aru Arib
Arbi Arbb + B
(3)
The non-zero components of the force fpi G (D” in eq. (2) of the inner part
of the rotor are the components of the unbalance force /y, i.e. introducing
the control matrix Su € ^ dynamic stiffness
matrix Ar of the rotor is given in terms of the matrices of inertia and
stiffness which are defined by modal analysis and by its physical data given
by the manufacturer. Each of the m shafts of the rotor is connected to
the foundation usually via 2 journal bearings. Since the dynamic stiffness
matrix B of the journal bearings represents a model for the oil film it consists
of connecting dof only. It can be shown that B is block diagonal
B =
0
0
(4)
containing the dynamic stiffness matrices Bi = Ki+ jujDi.i = 1, • • • ,m, of
the m bearings. Ki, Di are the matrices of stiffness and damping respectively
1028
which result from linearisation and are in general non-symmetric and non¬
singular. Eq. (1) is then used to estimate the foundation transfer function
F{u). This has been discussed in several papers ([1],[2],[3],[4],[5]). Lees et
al. [3] pointed out that fpB is sensitive with respect to deviations in the
model of the bearings over part of the frequency range. This sensitivity is
transfered to the model estimates.
In this paper a method is introduced which enables the decoupling of the
two problems of model estimation and of sensitivity of the foundation model
with respect to the model of the bearings. The basic idea of this method is
to estimate a transfer function H{u) which maps the displacements ufb{^)
to a force /^(w) rather than the force i-e.
H(u)ufb{^) = (5)
In extension of the earlier method the force //f(a;) can be chosen to be of
minimum sensitivity with respect to the model of the bearings. This robust
estimated transfer function H{uj) is related to that of the foundation F(uj)
by a transformation P{u)
H{lo) = PMF(a;), (6)
which of course retains the sensitivity with respect to the model of the
bearings. But since P{uj) only depends on the models of the rotor and the
bearings in the case of a modification within the model of the bearings no
new model estimation has to be performed because this has been done ro¬
bustly with respect to such model changes.
2 THE OPTIMUM CHOICE OF THE
FORCE VECTOR
As stated in Lees et al. [3] the sensitivity of the force /fb with respect to B
is mainly due to the inversion of the matrix Arb in eq. (2). It can be shown
that the condensation method of estimating the force /fb results from the
special case of eliminating the last 4m rows of the matrix
W :=
Arii
^RBI
0
Arib
ArbB -b P
^ ^(n+8m.)x(n+4m)
(7)
which can be written as Arb = T'^W ^ (l^(n+4m.)x(n+4m) defining the
selecting matrix of the master dof as
T := [ei, • • • , € R(-+8n)x(n+4m)^ (S)
1029
where in general en denotes a unit vector of appropriate dimension contain¬
ing zeros everywhere but in the nth place. In extension to the force fpB
defined in eq. (2) for an arbitrary selecting matrix T € the
condensation leads to a force fn given by
j-^LT
In+Sm-W{T'^Wy'T^
” *■ V*' . . .
=:P
(
Bufb
V }
( ^ \
0
V -^4m J
fpB-
(9)
Here T-^ € denotes the matrix which selects the slave dof,
and in general In denotes the unit matrix of dimension n. Indeed, inserting
the special choice of T from eq. (8) into eq. (9) leads to the sensitive force
Ih = fpB as defined in eq, (2).
The reason for the sensitivity of /fb is that the subsystem of the rotor
has low damping. Near the resonance frequencies of Arb its large condition
number depends sensitively on B. Thus the sensitivity of Jfb with respect
to B is due to a large condition number of Arb • Let T denote the set of all
possible selecting matrices, i.e.
r := {[ei..--',ew4„]:e4elR"+*“.
l<4<n + 8m, V /c = 1, • • « ,n + 4m}, (10)
One criterion for an optimum choice of the force Jr may be formulated as
the following minimisation problem:
Criterion 1:
The optimum choice is the solution of
mmcond{T^W), (11)
where W is defined in eg. (7).
A low condition number is necessary but not sufficient in order to provide
a low sensitivity of the force /h* Therefore a numerical test can be applied
using stochastic deviations in the bearing model. Let A A consist of
uniform distributed non-correlated random numbers with zero mean values
and variances equal to 1/3 for alH G 1, ■ • * , 2m. Define
ABi = ABfisi, Ti) := siAKi -h A A, (12)
where the positive scalars si.Ti control the magnitude of the random error
of the i-th bearing model. Thus, the error AB = AB{s,r) of the bear¬
ing model is well defined for s := (si, • ■ • , 52771)"^ and r := (ri, • • • ,r2mV ■
1030
Regard the force fn = f{uj,T,AB) as a function of the selecting ma¬
trix T and the bearing model error AB. For I random samples AB{k) =
AB(r{k),s{k)),k == calculate for each frequency a; G the up¬
per and lower bounds for the real and imaginary part of each component
/i, ^ = 1, • • • , 4m of the force vector /, i.e.:
//Lax(“.U
;= max Re {/i(a;,T, AB(/c))} ,
(13)
:= min Re {fi{u,T, AB{k))} ,
(14)
:= ^max^Im {/i(a;,T, AB(/c))} ,
(15)
fLni^.T)
:= min Im AB{k))} .
k=l,"-,l
(16)
Defining the force vectors
/max(w,T)
■■= T)+j- T),
(17)
/mm(w,T)
(18)
the second criterion can be formulated as a minimax problem:
Criterion 2:
The optimum selection is obtained from
4m
minmax^l/imax(w,r) - (19)
TST uj£u . -
2 = 1
Before the method outlined is demonstrated by an example some aspects of
the mathematical model of the foundation and methods for its estimation
based on the input/output equation (5) will now be considered.
3 ESTIMATION OF THE FOUNDATION
MODEL
The purpose of this section is to estimate the unknown foundation model
represented by the matrix
F{uj) = Afbb{^) - Afbi{^)Af]j{uj)Afib{^)‘ (20)
This expression results from dynamic condensation of the dynamic stiffness
matrix of the foundation
Af
Afbb Afib
Afib Afh
(21)
1031
which is partitioned with respect to its inner dof (index I) and those dof
coupled to the bearings (index B). For viscous damped linear elastomechan-
ical models the dynamic stiffness matrix Aj?(a;) of the foundation is given
by ^
Af(uj) := (22)
i=0
The matrices Ai are real valued and represent the contributions of stijffness,
damping and inertia for z = 0, 1, 2 respectively. In this case the identifica¬
tion of the foundation model requires the estimation of the three matrices
Ai which are parameterised by introducing dimensionless adjustment pa¬
rameters aik € IR, for all /c = 1, * • • , ^ = 0, 1, 2 (see for instance [6] or
[7] ). Those parameters are related to given real- valued matrices Sik by
Ni
Ai{ai)
Jk=i
Writing the adjustment parameters as one vector a"’’ ;= (af,aj’,aj) G IR^,
p := ATq + Ni + W2, the estimation of the foundation model is equivalent to
the estimation of the parameter vector a. The dynamic stiffness matrix of
the foundation becomes a nonlinear function of this parameter vector
^"(0;) =F[(jj^a) — Afbb[^-)0) “ Afb7(ci;j a)Ap}j(a;, a)Af’/5(a;, a). (24)
Substituting the measured quantities for upB and fu into eqs. (5) and
(9) the parameter vector a is usually estimated by minimising some norm of
the difference between measured and calculated quantities, called residuals
[8] . Using equation (5) is equivalent to the input residual method. Defining
the ith partial input residual as
(25)
where the dependency on the model parameters a of the input vector is
defined by
f{u), a) := P{uj)F[u), a)uFB(w), (26)
the cost function to be minimised is given by
M
JKa):=£^;}(i)W,(i)^/(i), (27)
i=l
where Wj{i) represents a weighting matrix for the zth partial residual and
the superscript f denotes the conjugate- transpose. The inverse problem (27)
is nonlinear with respect to the parameters to be estimated. Thus, there
is no advantage relative to the output residual method. Defining the ith
partial output residual as
vo{i) := u{ui, a) - (28)
1032
where the model output is defined by
(29)
0
0
ApiiiuJ^a) .
(30)
(31)
Woii) denotes a weighting matrix for the ith partial output residual.
Mathematical modelling is always purpose orientated [9]. In the case
discussed in this paper the purpose is to estimate the influence of the foun¬
dation on the dynamics of the rotor. For this purpose, no physically inter¬
pretable model is necessary in order to model this influence. In the next
section an alternative mathematical model is introduced which leads to a
linear inverse problem.
3.1 ARMA MODELLING IN THE FREQUENCY
DOMAIN: THE FILTER MODEL
Auto Regressive Moving Average models are well developed (see for in¬
stance [10], [7]) in order to simulate dynamic system behaviour. ARMA
models are defined in the time domain by
- lAt) = - ^A^), (32)
i=0 i=0
where the present output (state or displacement) u{t) due to the present
input f{t) depends on rio past outputs and on rii past inputs.
In the frequency domain eq. (32) leads to a (frequency-) filter model [7].
With reference to eq. (5) it has the form
Su
0
0
0
u(w,a) := [0,0,/2(m+i),0].4 ^(w,a)
with the dynamic stiffness matrix A of the entire model
A{u, a) =
Arii{u) Arib{^^) 0
Arbi{^) Arbb{‘^) B{lo) —B(u)
0 —B{uj) R(a;) 4- Afbs(^) g)
0 0 Afib{^jO,)
the cost function to be minimised is
M
Jo(a) :=E^oW^oW«o(i).
The output and input powers Uo^rii respectively, and the matrices
(•SA:)fc=o,-,ni are called filter^parameters and Jiave to be esti¬
mated. Of course the minimum of det[A(a;)] and of det[B(cj)] correspond
to the resonance and anti-resonance frequencies of the subsystem of the
foundation respectively.
For an optimum choice of P (see eq. (9)) the estimation of A and B can
be considered to be independent of the precise values of the model of the
bearings. Thus, the problem of the uncertainty in the bearmg models and
the problem of model estimation are decoupled. If A and B are estimated
refering to eq. (5) then
= H. (34)
The estimation of the filter parameters is robust with respect to deviations
in the bearings model. Thus, the uncertainty of the estimation of the foun¬
dation model F is due to the inversion of the matrix P only
F = P-^H = p-'-B~^A, (35)
which represents a problem a priori and which occurs only in the calculation
of the force of the foundation at the bearings
/PB = P-^B~^Aufb- (36)
Of course the force vector fpB is sensitive to changes in the bearings model
but only due to corresponding changes in P. The estimated part B A is
robust with respect to changes in the bearings model.
In order to calculate the response ufb iio explicit calculation of the
inverse of P is necessary,
upB = A Bfn- (37)
Since the estimated model and the force Jh are insensitive with respect to
the bearings model the estimation otupB is robust in this sense.
Of course the influence of errors in up b and fu have not yet been taken
into account. Accordingly the model powers n^n^, must be estimated as
well as the matrices Ai.Bk- The estimation method is outlined in the fol¬
lowing section.
3.2 ESTIMATION OF THE FILTER
PARAMETERS
In order to estimate the filter parameters the least squares method can be
applied to minimise the equation error in eq. (33). Defining the zth partial
equation residual as
vsij) '= A[ui)upB[^i) - B{uJi)fH{^i) (38)
1034
the cost function to be minimized is given by
M
i=l
(39)
where W^ii) denotes a weighting matrix for the ith. equation residual. As¬
suming WeIi) = Am for alH = 1, • • ’ , M, the filter equation (33) can be
extended for M excitation frequencies as
= (40)
where C/, 2^ and A are defined by
U
Z
A
• • • 5 ,
r a;i 0 1
lom
(41)
(42)
(43)
The solution of the minimisation problem (39) is equivalent to the normal
solution of eq. (40) which can be rewritten as
[Afio 1 ' ' ' 1 -Ao ) ‘ : ■^o]
=: V
U
Z
= 0
(44)
Because the filter parameters represented by the matrix V €
j^4mx4m(no+ni+2) leal-valued, equation (44) must be satisfied for the
real and imaginary parts of the matrix Y € which finally
yields
1/ [Re {Y} , Im {Y}] =: VA = 0. (45)
This problem does not lead to a unique solution for the filter parameters.
Indeed, for any arbitrary non-singular matrix C
CAufb = (46)
is also a solution. But since one is interested (see eq. (34)) in the product
B~^ A (or its inverse) only this final result is of interest and this product is
unique.
1035
As a necessary and a sufficient condition for a full-rank solution V of
eq. (45) the matrix X € 5^4m(no+ni4-2)x2M a rank deficiency of
4m, i.e.
rank(A) = 4m (no + n^ + 1). (47)
Of course this problem has to be treated numerically. The rank decision
is usually made by looking to the singular values 7(no, rii) 6 5^4m(no+ni+2)
of the matrix X = X{no,ni). Because one cannot expect to achieve zero
rather than relative small singular values one has to define a cut-off limit.
This is due to the fact that the equation error (39) can be made arbitrary
small by increasing the degree p := no + Ui of the filter model. The same
situation occurs if one looks at the maximum relative input error
6/ :=
max
(48)
or to the maximum relative output error
en :== max
II^fs(^2)1I
(49)
With increasing degree p the errors ej and cq can be made arbitrary small.
This is a typical expression for an ill-posed problem which can be turned
into an well-posed problem by applying regularisation methods [11]. To
choose an appropriate regularisation method needs further investigation and
is beyond the scope of this paper. In the next section the method of choosing
the optimum force vector fn is demonstrated by a simple example.
4 A SIMPLE EXAMPLE
The test model is depicted in Fig. 1. The one-shaft-rotor is simulated by
an Euler-Bernoulli beam which is spatially discretised with 10 dof. Accord¬
ing to the partition with respect to inner points and interface points (see
eq. (3)) the number n of inner rotor dof is 8 and the number of connecting
dof is 2. Only one translation dof of the rotor is connected to each bearing
which are modelled by massless springs with stiffnesses ki — 1.77 • 10® and
k2 = 3-54 • 10® N/m respectively. The foundation is modelled by an uncon¬
nected pair of masses mi = 90, m2 = 135 kg and springs with stiffnesses
kfi = /c/2 = 1.77 • 10® N/m. The force fu due to an unbalance 6 = 0.01
kg-m is given by fu{<^) ■= € IR. The force vector f^i e IR® is assumed
to have one non-zero component only, i.e. Jri fuc^. The frequency range
between 0 and 250 Hz is discretised with equally spaced stepsize of 0.5 Hz.
The selecting matrix T € of the master dof is assumed to consist of
the unit vector 64 in order to select out the unbalance force fjj because this
1036
Figure 1: The simple test model
excitation is independent of the model of the bearings and therefore of min¬
imum sensitivity. Thus the remaining redundancy consists in eliminating
one row of the matrix
G := [ei, 82. 83. 65. • • • . eul'' W € (50)
The result of the first criterion are depicted in Fig. 2. It shows the frequency
dependent condition numbers for the elimination of each row of the matrix
G in turn. This leads to an optimum choice by eliminating the 4-th row of
G. Thus, the optimum choice of the master dof is given by T*^ = [64,65].
This result is confirmed by applying the second optimisation criterion.
For this purpose a uniform distributed uncorrelated random error with
zero mean value is added to the stiffnesses of the bearings simulating a
model variation of 50 per cent, i.e. ki — ^ ki -1- Nzikil2, where =
1,2, are uncorrelated random numbers with expectation value E{Aki} = 0
and with variance E{AkiAkj} = For a size of I - 500 random
samples, the upper and lower bounds /max, /min and /f max, /h min of the
force fpB = /fb(<^, A5) and of the force Jh = /(w,T, AB) respectively
with the selecting matrix T - [64,65], have been calculated. In contrast to
the maximum difference of upper and lower bounds of the force f^B of ~ 20,
that of the force fn is of the order of the computational accuracy ~ 10"'^,
and is therefore negligible. In a first step the force /fb is used to estimate
the filter parameters of the model F of the foundation. Solving the singular
value decomposition for all input and output powers {ni.rio) 6 [0, 5]^ C
the calculation of the maximum relative input and of the maximum output
errors as defined by eqs. (48) and (49) with fn ~ Ifb leads to the results
1037
Elimination of row number...
Frequency [Hz]
Frequency [Hz]
Figure 2: Frequency dependent condition numbers
depicted in Fig. 3. For a model realisation with [ni.rio) = (0,2) the values
of the maximum relative errors are approximately ej 7.1 • and eo ~
9.0-10“”^ which corresponds to the computational accuracy. Using this model
the estimates of the filter matrices correspond within the computational
accuracy to those of the ‘true’ foundation model.
Using variations of the force fpB between the bounds /max, /min froni
the second criterion the associated upper and lower bounds of the relative
input and output error have been calculated. The influence of the variation
1038
Firgure 3: Maximum relative input and output error for different
input powers rii and output powers rio using the force vector fpB
of the bearing stiffness of 50% leads via the associated variation of the
force fpB to drastic variations of the relative input and output errors. The
difference of upper and lower bound of the maximum relative input error is
1039
Output power
Figure 4: Maximum relative input and output error for different
input powers rii and output powers Uo using the optimised force
vector Jh
of order 100 and that of the maximum relative output error is approximately
LOl
The situation is different using the optimised force fn in order to es¬
timate the filter parameters of the model H. For each input and output
power {ni,no) e [0,25]^ C the maximum relative input error and the
maximum relative output error have been calculated. The result is shown in
Fig. 4. For a maximum relative input error e; 6.5 • 10“® a filter model of
degree 12 is available with the powers (n^, = (4, 8). This model produces
a maximum relative output error eo ^ 10“®.
Analogous to the robustness investigations for fpB now for the force
/h the upper and lower bounds of the relative input and of the relative
output error due to the random variation in the bearing models have been
calculated. For the chosen model with powers (nj,no) = (4, 8) the difference
of upper and lower bound of the relative input error as well that of the
relative output error are of about the same order 10“^. Thus, compared
with the order of variation 100 and 10"^ of the direct foundation model
estimate the estimate of H is robust with respect to changes in the bearing
model.
5 CONCLUSION
In this paper a method is introduced which enables the decoupling of the
two problems of model estimation and of sensitivity of the foundation model
with respect to the model of the bearings. The method produces an opti¬
mised choice of the input /output equation which provides a transfer func¬
tion estimation that is robust with respect to deviations in the model of
the bearings. For the foundation model estimation a filter model is intro¬
duced. This modelling strategy has the advantage of leading to a linear
inverse problem. The disadvantage is that with increasing model degree
the equation error can be made arbitrarily small. Because this error should
not become smaller than the accuracy of the data, a cut-off limit has to be
determined a priori. Further investigations should allow the cut-off limit to
be related to the data errors.
REFERENCES
1. Feng, N.S. and Hahn, E.J., Including Foundation Effects on the Vi¬
bration Behaviour of Rotating Machinery. Mechanical Systems and
Signal Processing, 1995 Vol. 9, No. 3, pp. 243-256.
2. Friswell, M.I., Lees, A.W. and Smart, M.G., Model Updating Tech¬
niques Applied to Turbo-Generators Mounted on Flexible Founda¬
tions. NAFEMS Second International Conference: Structural Dynam¬
ics Modelling Test, Analysis and Correlation, Glasgow: NAFEMS,
1996 pp. 461-472.
1041
3. Lees, A.W. and Friswell, M.L, Estimation of Forces Exerted on Ma¬
chine Foundations. Identification in Engineering Systems, Wiltshire;
The Cromwell Press Ltd., 1996, pp. 793-803.
4. Smart, M.G,, Friswell, M.L, Lees, A.W. and Prells, U., Errors in
estimating turbo-generator foundation parameters. In Proceedings
ISMA21 - Noise and Vibration Engineering, ed. P. Sas, Katholieke
Universiteit Leuven, Belgium, 1996, Vol. II, pp. 1225-1235
5. Zanetta, G.A., Identification Methods in the Dynamics of Turbogener¬
ator Rotors. The International Conference on Vibrations in Rotating
Machinery, IMechE, C432/092, 1992, pp. 173-181.
6. Friswell, M.L and Mottershead, J.E., Finite Element Model Updating
in Structural Dynamics. Dordrecht, Boston, London: Kluwer Aca¬
demic Publishers, 1995.
7. Natke, H.G., Einfuhrung in die Theorie und Praxis der Zeitreihen-
und Modalanalyse - Identifikation schwingungsfdhiger elastomechanis-
cher Systeme. Braunschweig, Wiesbaden: Friedrich Vieweg Sz Sohn,
1993.
8. Natke, H.G., Lallement, G., Cottin, N. and Prells, U., Properties of
Various Residuals within Updating of Mathematical Models. Inverse
Problems in Engineering, Vol. 1, 1995, pp. 329-348.
9. Natke, H.G., What is a true mathematical model? - A discussion of
system and model definitions. Inverse Problems in Engineering, 1995,
Vol. 1, pp. 267-272.
10. Gawronski, W. and Natke, H.G., On ARMA Models for Vibrating
Systems. Probabilistic Engineering Mechanics, 1986, Vol. 1, No. 3,
pp. 150-156.
11. Baumeister,J., Stable Solution of Inverse Problems. Braunschweig,
Wiesbaden: Friedrich Vieweg &; Sohn, 1987.
ACKNOWLEDGEMENTS
The authors are indebted to Nuclear Electric Ltd and Magnox Pic for fund¬
ing the research project BB/G/40068/A to develop methods which enables
the estimation of the influence of the foundation on the dynamics of the
rotor. Dr. Friswell gratefully acknowledges the support of the Engineering
and Physical Sciences Research Council through the award of an Advanced
Fellowship.
1042
MATHEMATICAL HYSTERESIS MODELS AND
THEIR APPLICATION TO NONLINEAR
ISOLATION SYSTEMS
Y.Q. Ni, J.M. Ko and C.W. Wong
Department of Civil and Structural Engineering
The Hong Kong Polytechnic University, Hong, Kong
Abstract
Two mathematical hysteresis models, the Duhem-Madelung (DM)
model and the Preisach model, are introduced to represent the
hysteretic behavior inherent in nonlinear damping devices. The DM
model generates the hysteresis with local memory. Making use of the
Duhem operator, the constitutive relation can be described by single¬
valued functions with two variables in transformed state variable
spaces. This makes it feasible to apply the force-surface nonparametric
identification technique to hysteretic systems. The Preisach model can
represent the hysteresis with nonlocal memory. It is particularly
suitable for describing the selective-memory hysteresis which appears
in some friction-type isolators. An accurate frequency-domain method
is developed for analyzing the periodic forced vibration of hysteretic
isolation systems defined by these models. A case study of wire-cable
vibration isolator is illustrated.
1 . Introduction
The dynamic response of a structure is highly dependent on
the ability of its members and connections to dissipate energy by
means of hysteretic behavior. The assessment of this behavior
can be done by means of experimental tests and the use of
analytical models that take into account the main characteristics
of this nonlinear mechanism. Although a variety of hysteresis
models have been proposed in the past decades, many structural
systems exhibit more complicated hysteretic performance (mainly
due to stiffness or/ and strength degrading) which the models in
existence are reluctant and even inapplicable to depict [1,2]. On
the other hand, nonlinear vibration isolation has recently been
recognized as one of effective vibration control techniques. In
1043
particular, hysteretic isolation devices have got wide applications
owing to their good diy friction damping performance. These
hysteretic isolators may exhibit very complicated features such
as asymmetric hysteresis, soft-hardening hysteresis, nonlocal
selective-memory hysteresis [3-5]. None of the models available
currently in structural and mechanical areas can represent all
these hysteresis characteristics. Other more elaborate hysteresis
models need to be established for this purpose.
In reality, hysteresis phenomenon occurs in many different
areas of science, and has been attracting the attention of many
investigators for a long time. However, the true meaning of
hysteresis varies from one area to another due to lack of a
stringent mathematical definition of hysteresis. Fortunately,
because of the applicative interest and obvious importance of
hysteresis phenomenon, Russian mathematicians in 1970’s and
the Western mathematicians in 1980’s, began to study hysteresis
systematically as a new field of mathematical research [6,7]. They
also deal with the hysteresis models proposed by physicists and
engineers in various areas, but they separate these models from
their physical meanings and formulate them in a purely
mathematical form by introducing the concept of hysteresis
operators. Such mathematical exposition and treatment can
generalize a specific model from a particular area as a general
mathematical model which is applicable to the description of
hysteresis in other areas. In this paper, two mathematical
hysteresis models are introduced and the related problems such
as identification and response analysis encountered in their
application to nonlinear isolation systems are addressed.
2. Definition of Hysteresis
Hysteresis loops give the most direct indication of hysteresis
phenomena. But it is intended here to introduce a mathematical
definition of hysteresis. Let us consider a constitutive law: u r,
which relates an input variable u(t) and an output variable r\t).
For a structural or mechanical system, u(t) denotes displacement
(strain); r{t) represents restoring force (stress); t is time. We can
define hysteresis as a special type of memoiy-based relation
1044
between u(t) and r(t). It appears when the output r[t) is not
uniquely determined by the input u(t) at the same instant t, but
instead r(t) depends on the evolution of u in the interval [0, t] and
possibly also on the initial value ro, i.e.
r(t) = iR[u(-),ro](t) (1)
where the memory-based functional iR[u(-),ro](t) is referred to as
hysteresis operator. In order to exclude viscosity-type memory
such as those represented by time convolution, we require that 91
is rate-independent, i.e. that r(t) depends just on the range of u in
[0, t] and on the order in which values have been attained, not on
its velocity. In reality, memory effects may be not purely rate
independent as hysteresis is coupled with viscosity- type effects.
However, as shown later, in most cases the rate independent
feature of hysteresis is consistent with experimental findings,
especially when evolution (variation in time) is not too fast.
3. Duhem-Madelung (DM) Model
3.1 Formulation
The DM model can be defined with or without referring to a
confined hysteresis region. For the structural or mechanical
hysteretic systems, it is not necessary to introduce the notion of
bounded curves because there exists neither the saturation state
nor the major loop. In this instance, the DM model establishes a
mapping (named Duhem operator) 91: (u, ro) rby postulating the
following Cauchy problem [7,8]
= g, (li. '■) ■ w, (t) - 02 (u, r) • u_ (t) (2a)
r(0) = r„ (2b)
where an overdot denotes the derivative with respect to t; gi{u,r]
and g2{ii,r) are referred to as ascending and descending functions
(curves) respectively; and
tijt) = max[0,u(t)] = t [|u(t)| + u(t)] (3a)
u.(t) = min[0,u(t)] = ^[|u(t)| - u(f)] (3b)
Eq. (2a) can be rewritten as
f(f) = p[u,r,sgn(u)]-u(t) (4)
1045
u> 0
u<0
(5)
in which the describing function has the form
g{u,r,sgn{u)]
fg{u,r,l) = g,{u,r)
[g[u,r-l) = g^(u,r)
It is obvious that the DM model is rate independent. In
addition, it is specially noted that in this constitutive law the
output r{t) is not directly dependent on the entire history of u{t)
through [0,t]; but instead depends only on the local histoiy
covered since the last change of sgn(u) and on the value of the
output at this switching instant. It means that the output can
only change its character when the input changes direction. As a
consequence, the DM model usually represents the hysteresis
with local memory except that the functions gi(u,?) and g2{u,7) are
re-specified as hysteresis operators.
Within the framework of DM formulation, the ascending and
descending functions gi(u,T] and g2(u,r) are just required to fulfill
suitable regularity conditions and need not to be specified in
specific expressions, so both the form and parameters of the
functions can be fine-tuned to match experimental findings. On
the other hand, the DM formulation can deduce a wide kind of
differential-type hysteresis models such as Bouc-Wen model,
Ozdemir’s model, Yar-Hammond bilinear model and Dahfs
frictional model. For the Bouc-Wen model
r(t) = K.u{t) + z(t) (6a)
z{t) = au(t] - P|ti(f)|z(f)|z(t)|'”‘ - YU(t)|z(f )|" (6b)
it corresponds to the DM model with the specific ascending and
descending functions as
grj(u,r) = a + K “ [y + p sgn(r -Ku)]|r -Kup (7a)
g2(w,r) = a + K - [y - p sgn(r -Kw)]|r -ku|” (7b)
and for the Yar-Hammond bilinear model
f{t) = {a - y sgn(ii) sgn[r - p sgn(ii)]}ii (8)
its describing function is independent of u(t) as follows
5f[u,r,sgn(u)l = gf[r,sgn(u)l = a -y sgn(u)sgn[r - p sgn(u)] (9)
Hence, the Duhem operator also provides an accessible way
to construct novel hysteresis models by prescribing specific
1046
expressions of the ascending and descending functions.
Following this approach, it is possible to formulate some models
which allow the description of special hysteretic characteristics
observed in experiments, such as soft-hardening hysteresis,
hardening hysteresis with overlapping loading envelope, and
asymmetric hysteresis [5].
3.2 Identification
System identification techniques are classified as parametric
and nonparametric procedures. The parametric identification
requires that the structure of system model is a priori known. The
advantage of nonparametric identification methods is that they
do not require a priori the knowledge of system model. The most
used nonparametric procedure for nonlinear systems is the force
mapping (or called force surface) method [9]. This method is
based on the use of polynomial approximation of nonlinear
restoring force in terms of two variables — ^the displacement u(t)
and the velocity u{t) . For nonlinear hysteretic systems, however,
the hysteretic restoring force appears as a multivalued function
with respect to the variables u{t) and u(t) due to its history-
dependent and non-holonomic nature. This renders the force
mapping method inapplicable to hysteretic systems, although
some efforts have been made to reduce the multivaluedness of
the force surface [10,11].
One of the appealing virtues of the DM model is that it can
circumvent this difficulty. Making use of the Duhem operator,
the hysteretic constitutive relation of Eq.(l) is described by two
continuous, single-valued functions gi(u,r) and g2{u,r) in terms of
the displacement u(t) and the restoring force r{t). Thus, single¬
valued “force” surfaces gi{u,r) and g2{u,7) can be formulated in
the subspaces of the state variables {u,r,gi) and (u,r,g2), and can
be identified by using the force mapping technique. Following
this formulation, a nonparametric identification method is
developed by the authors [12]. In this method, the functions
g\(u,7) and g2{u,7) are expressed in terms of shifted generalized
orthogonal polynomials with respect to u and r as follows
gM,r) = i = 0’'(u)G<''0(r) (10a)
i=lj=l
1047
(10b)
g^r] = z i(|>,.(r)gf(i>,(u) = 0^(u)G'^'<l>(r)
I=lj=l
where = [gf ]^xn =[5'zf]mxn ^re called the expansion-
coefficient matrices of gi{u,r) and g2{u,T]. Some algorithms have
been proposed to estimate the values of these coefficient matrices
based on experimentally observed input and output data. It
should be noted that here the vectors 0(u) and <^{r) are shifted
generalized orthogonal polynomials [13]. They are formulated on
the basis of common recurrence relations and orthogonal rule,
and cover all kinds of individual orthogonal polynomials as well
as non- orthogonal Taylor series. Consequently, they can obtain
specific polynomial-approximation solutions of the same. problem
in terms of Chebyshev, Legendre, Laguerre, Jacobi, Hermite and
Ultraspherical polynomials and Taylor series as special cases.
4. Preisach Model
4. 1 Formulation
The intent of introducing the Preisach model is to supply the
lack of a suitable hysteresis model in structural and mechanical
areas, which is both capable of representing nonlocal hysteresis
and mathematically tractable. Experiments revealed that the
hysteretic restoring force of some cable-type vibration isolators
relates mainly to the peak displacements incurred by them in the
past deformation [3]. It will be shown that the Preisach model is
especially effective in representing such nonlocal but selective-
memory hysteresis, in which only some past input extrema (not
the entire input variations) leave their marks upon future states
of hysteresis nonlinearities.
The Preisach model is constructed as a superposition of a
continuous family of elementary rectangular loops, called relay
hysteresis operators as shown in Fig. 1. That is [7,14],
r(f) = 'R[u(-)](f) = j||^(a,P)y„D[u(t)]dadf5 (11)
a>P
where ia(a,|3) > 0 is a weight function, usually with support on a
bounded set in the (a,p)-plane, named Preisach plane; Ya.p[^(^)] is
the relay hysteresis operator with thresholds a > p. Outputs of
1048
these simplest hysteresis operators have only two values +1 and
-1, so can be interpreted as two-position relays with “up” and
“down” positions corresponding to ya,p[^^(^)l=+l and
1
+1
p
a
-1
Fig. 1 Relay Hysteresis Operator
Hence, the Preisach model of Eq.(ll) can be interpreted as a
spectral decomposition of the complicated hysteresis operator iR,
that usually has nonlocal memory, into the simplest hysteresis
operators 7ct,p with local memory. In the following, we illustrate
how the model depicts the nonlocal selective-memory feature.
Consider a triangle T in the half-plane a > (3 as shown in
Fig.2. It is assumed that the weight function )i(a,p) is confined in
the triangle T, i.e. }i(a,p) is equal to zero outside T. Following the
Preisach formulation, at any time instant t, the triangle T can be
subdivided into two sets: S^(t) consisting of points (a,P) for which
the corresponding Yc^.p-operators are in the “up” position; and S^(t)
consisting of points {a,P) for which the corresponding Ya.p“
operators are in the “down” position. The interface L(t) between
S^(t) and Sr(t) is a staircase line whose vertices have a and p
coordinates coinciding respectively with local maxima and
minima of input at previous instants of time. The nonlocal
selective-memory is stored in this way. Thus, the output r(^) a-t
any instant t can be expressed equivalently as [14]
r{t)= J|^(a,P)dadp- |jM,(a,P)dadp (12)
S'(£) S'(t)
It should be noted that the Preisach model does not
accumulate all past extremum values of input. Some of them can
be wiped out by sequent input variations following the wiping-out
property {deletion rule): each local input maximum wipes out the
1049
vertices of L(t) whose a-coordinates are below this maximum, and
each local minimum wipes out the vertices whose p-coordinates
are above this minimum. In other words, only the alternating
series of dominant input extrema are stored by the Preisach
model; all other input extrema are erased.
Fig. 2 Input Sequence and Preisach Plane
4.2 Identification
It is seen from Eq.(l 1) that the Preisach model is governed by
the weight function |i(a,p) after determining L(t) which depends
on the input sequence. !i(a,P) is a single-valued function with
respect to two variables a and p. Hence, the aforementioned
nonparametric identification method can be also implemented to
identify iLL(a,p) by expanding it in a similar expression to Eq.(lO).
An alternating approach is to define the following function
H'(a',P') = |j|a(a,p)dadp = ^.[^ ^(ot,P)da]dp (13)
r(cc'.3')
where T(a',p') is the triangle formed by the intersection of the
line a = a' , p = p' and a = p . Differentiating Eq.(13) yields
^(a',P') = --
aa'ap'
(14)
Thus, the force mapping identification technique can be applied
to determine H(a,p) consistent with the experimental data, and
then ^(a,p) is obtained by Eq.(14).
5. Steady-State Response Analysis
Hysteretic systems are strongly nonlinear. A study of the
steady- state oscillation is one of the classical problems of
1050
nonlinear systems. Usually, the dynamic behavior of a nonlinear
system is represented by its resonant frequency and frequency
response characteristics. In the following, an accurate frequency-
domain method accommodating multiple harmonics is developed
to analyze the periodically forced response of hysteretic systems
defined by mathematical hysteresis models.
Fig. 3 shows a single-degree-of- freedom hysteretic oscillator
with mass m, viscous damping coefficient c, and linear stiffness
ky subjected to an external excitation F(t), for which the governing
equation of motion is
m • u(t) + c • u(t) + k ' u[t) + r{t) = F(t) (15)
where the hysteretic restoring force r(t) is represented by the DM
model as Eq.(4). It is worth noting that for the kinetic equation
Eq.(15), the excitation is F(t) and the response is u(t); and for the
hysteretic constitutive law Eq.(4), u(t) is input and r{t) is output.
The causal relationship is different.
Fig. 3 Single -Degree-of-Freedom Hysteretic System
Due to the hereditary nature of the hysteresis model, it is
difficult to directly solve the kinetic equation Eq.(15) by iteration.
Here, Eq.(15) is only used to establish the relation between the
harmonic components of u(t) and r(t). Suppose that the system is
subjected to a general periodic excitation F[t) with known
harmonic components F={Fq F^ F2 ••• F2 . The multi¬
harmonic steady-state response can be expressed as
a ^ ^ .
u{t) = ~ + a j cos j(Dt + a j sin jcot (16)
2 j=i
1051
in which a={ao a2 ••• a^v ^2 unknown
vector containing the harmonic components of u(t). Introducing
Eq.(16) into Eq.(15) and using the Galerkin method provide
ro^F^-k-a^ (17a)
Tj = Fj -c-citj-a] [j = 1,2, ••• , N) (17b)
Tj =Fj +c-coj-a^. -(/c-m-coV^)-a* (j = 1, 2, ,N) (17c)
where r={ro q ••• r^}'^ is the harmonic vector of
the hysteretic restoring force r(t). Referring to the hysteretic
constitutive law, we define the determining equation as
D(t) = r{t) - g[u,r,sgn(u)] • u{t) (18)
When a is the solution of u(t), applying the Galerkin method
into Eq.(18) and considering Eq.(17) achieve
d{a) = 0 (19)
where the vector d{a)={dQ ^3 --dj^ d^ d^ "• is comprised
of the harmonic components of D(t) corresponding to a. An
efficient procedure to seek the solution of Eq.(19) is the
Levenberg-Marquardt algorithm with the iteration formula
where the Jacobian matrix J[a(^)] = dd(a) / da\a=a^^) ; 9^ is the
Levenberg-Marquardt parameter and I is identity matrix.
At each iteration, the function vector and Jacobian
matrix should be recalculated with updated values of
Here, a frequency/ time domain alternation scheme by FFT is
introduced to evaluate the values of d(a) and J[a) at d{a)
and dd[a)/da are known to be the Fourier expansion coefficients
of D(t) and dD(t)/ da respectively. For a given a(^) and known F, the
corresponding r[a(^)] is obtained from Eq.(17), and the inverse
FFT is implemented for and r[a(^)] to obtain all the time
domain discrete values of u{t] , u(t) , r(t) and f(t) over an integral
period. Then the time domain discrete values of the function
D{u,u,r,r,t) , corresponding to a=a(^), are evaluated from Eq.(18).
Making use of forward FFT to these time domain discrete values
of D{u,u,r,r,t) , the values of function vector d[a(^)] are obtained.
1052
Similarly, the partial differential dD{t) / da can be analytically
evaluated in the time domain. Forward FFT to the time domain
values of dD(t) / da at ct=a(^) gives rise to dd[ai^)]/da.
6. Case Study
Wire-cable vibration isolators are typical hysteretic damping
devices. Dynamic tests show that their hysteresis behaviors are
almost independent of the frequency in the tested frequency
range [4,15]. Experimental study and parametric modelling of a
wire-cable isolator have been carried out [5]. Fig. 4 shows the
experimental hysteresis loops in shear mode. It is seen that for
relatively small deformations, the isolator exhibits softening
hysteresis loops. When large displacements are imposed, the
stiffness of the loops becomes smoothly hard. This nature is
referred to as soft-hardening hysteresis. Based on the Bouc-Wen
model, a parametric identification was performed to model these
hysteresis loops, but the result is unsatisfactory. This is due to
the fact that the Bouc-Wen model cannot represent such soft-
hardening nature of hysteresis.
Fig. 4 Experimental Hysteresis Loops
We now use the DM model to represent these hysteresis
loops, and perform a nonparametric identification to determine
the functions gi(u,7) and g2(u,7). The simplest Taylor series are
adopted, i.e. (i),(r) = (r / and (|)j(u) = (u j UqY~^ (ro = 20.0 and
uo = 2.0). Fig. 5 shows the identified “force” surfaces of gi{u,T) and
g2(u,r) by taking m=n~S. Fig. 6 presents the theoretical hysteresis
1053
loops generated by the DM model using the identified g\(u,T) and
g2(u,}). It is seen that the modeled hysteresis loops are agreeable
to the observed loops. In particular, the soft-hardening nature is
reflected in the modeled hysteresis loops.
-200
Fig. 6 Modeled Hysteresis Loops by DM Model
After performing the modelling of hysteretic behavior, the
dynamic responses of hysteretic systems can be predicted by the
developed method. Fig. 7 shows a vibration isolation system
installed with wire-cable isolators in shear mode. It is subjected
to harmonic ground acceleration excitation Xg[t)=Acos2Kft. The
equation of motion of the system is expressed as
m-u(t) + K ■r(t) = -rri' Xg(t) (21)
where m is the mass of the system; K is number of the isolators
installed. u{t) is the displacement of the system relative to the
ground. r(t) is the restoring force of each isolator and has been
determined from nonparametric identification.
Fig. 8 illustrates the predicted frequency-response curves of
the relative displacement when 7n=6kg and K=2. The excitation
amplitude A is taken as 0.25g, O.SOg, 0.35g, 0.40g and 0.45g
respectively. The frequency-response curves show clearly the
nonlinear nature of the wire-cable isolation system.
i
I
Xg(t)
Fig. 7 Vibration Isolation System with Wire -Cable Isolators
Fig. 8 Frequency Response Curves of Relative Displacement
1055
7. Concluding Remarks
This paper reports a preliminaiy work of introducing the
mathematical hysteresis models in structural and mechanical
areas. It is shown that a wide kind of differential hysteresis
models, which are extensively used at present, can be derived
from the Duhem-Madelung (DM) model. Thus, the mathematical
properties concerning the DM model are also possessed by these
models. Two potential advantages appear when the DM
formulation is used. Firstly, it allows to apply the force mapping
technique to hysteretic systems. Secondly, it provides an
approach to construct novel differential models which reflect
some special hysteretic characteristics. The Preisach model is
shown to be capable of representing nonlocal hysteresis and
mathematically tractable. It offers a more accurate description of
several observed hysteretic phenomena. Emphasis is placed on
demonstrating the selective-memory nature of this hysteresis
model. The case study based on experimental data of a wire-
cable isolator has shown the applicability of the mathematical
hysteresis model, and the validity of the steady-state response
analysis method proposed in the present paper.
Acknowledgment: This study was supported in part by the
Hong Kong Research Grants Council (RGC) and partly by The
Hong Kong Polytechnic University. These supports are gratefully
acknowledged.
References
1. Azevedo, J. and Calado, L., ‘‘Hysteretic behaviour of steel
members: analytical models and experimental tests”, J.
Construct Steel Research, 1994, 29, 71-94.
2. Kayvani, K. and Barzegar, F., “Hysteretic modelling of tubular
members and offshore platforms”, Eng. Struct, 1996, 18, 93-
101.
3. Lo, H.R., Hammond, J.K. and Sainsbury, M.G., “Nonlinear
system identification and modelling with application to an
isolator with hysteresis”, Proc. 6th Int modal Anal Conf.,
Kissimmee, Florida, 1988, Vol.II, 1453-1459.
1056
4. Demetriades, G.F., Constantinou, M.C. and Reinhorn, A.M.,
“Study of wire rope systems for seismic protection of
equipment in buildings”, Eng. Struct, 1993, 15, 321-334.
5. Ni, Y.Q., “Dynamic response and system identification of
nonlinear hysteretic systems”, PhD Dissertation, The Hong
Kong Polytechnic University, Hong, Kong, November 1996.
6. Krasnosefskii, M.A. and Pokrovskii, A.V., Systems with
Hysteresis, translated from the Russian by M. Niezgodka,
Springer-Verlag, Berlin, 1989.
7. Visintin, A., Differential Models of Hysteresis, Springer-Verlag,
Berlin, 1994,
8. Macki, J.W., Nistri, P. and Zecca, P., “Mathematical models
for hysteresis”, SIAM Review, 1993, 35, 94-123.
9. Masri, S.F. and Caughey, T.K., “A nonparametric
identification technique for nonlinear dynamic problems”, J.
Appl Mech, ASME, 1979, 46, 433-447.
10. Lo, H.R. and Hammond, J.K., “Nonlinear system identification
using the surface of nonlinearity form: discussion on
parameter estimation and some related problems”, Proc. 3rd
Int Conf Recent Adv. Struct. Dyn., Southampton, UK, 1988,
339-348.
11. Benedettini, F., Capecchi, D. and Vestroni, F., “Identification
of hysteretic oscillators under earthquake loading by
nonparametric models”, J. Eng. Mech., ASCE, 1995, 121, 606-
612.
12. Ni, Y.Q., Ko, J.M. and Wong, C.W., “Modelling and
identification of nonlinear hysteretic vibration isolators”.
Accepted to SPJE’s 4th Annual Symposium on Smart Structures
and Materials : Passive Damping and Isolation, 3-6 March
1997, San Diego, USA.
13. Ni, Y.Q., Wong, C.W. and Ko, J.M., “The generalized
orthogonal polynomial (GOP) method for the stability analysis
of periodic systems”, Proc. Int. Conf. Comput. Methods Struct.
Geotech. Eng., Hong Kong, 1994, Vol.II, 464-469.
14. Mayergoyz, I.D., Mathematical Models of Hysteresis, Springer-
Verlag, New York, 1991.
15. Ko, J.M., Ni, Y.Q. and Tian, Q.L., “Hysteretic behavior and
empirical modeling of a wire-cable vibration isolator”, Int. J.
Anal. Exp. Modal Anal, 1992, 7, 111-127.
1057
1058
The identification of turbogenerator foundation models
from run-down data
M Smart, M I Friswell, A W Lees, U Prells
Department of Mechanical Engineering
University of Wales Swansea, Swansea SA2 8PP UK
email: m.smart@swansea.ac.nk
ABSTRACT
The trend of placing turbines in modern power stations on flexi¬
ble steel foundations means that the foundations exert a considerable
influence on the dynamics of the system. In general, the complex¬
ity of the foundations means that models are not available a priori,
but rather need to be identified. One way of doing this is to use
the measured responses of the foundation at the bearings to the
synchronous excitation obtained when the rotor is run down. This
paper discusses the implementation of such an estimation technique,
based on an accurate model of the rotor and state of unbalance, and
some knowledge of the dynamics of the bearings. The effect of errors
in the bearing model and response measurements on the identified
parameters is considered, and the instrumental variable method is
suggested as one means of correcting them.
1 INTRODUCTION
The cost of failure of a typical turbine in a modern power station is
very high, and therefore development of condition monitoring techniques
for such machines is an active area of research. Condition monitoring relies
on measuring machine vibrations and using them to locate and quantify
faults, which obviously requires an accurate dynamic model of the ma¬
chine. Although the dynamic characteristics of rotors are generally well
understood, the foundations on which they rest are not. Since the founda¬
tions are often quite flexible, they can contribute considerably to the rotor’s
dynamic behaviour.
Finite element modelling has been attempted but the complexity of
the foundations, and the fact that they often differ substantially from the
original drawings rendered the technique generally unsuccessful [1]. Exper¬
imental modal analysis is another possible solution, but this requires that
the rotor be removed from the foundation, and that all casings remain in
place, which is not practical for existing power plant. However, mainte¬
nance procedures require that rotors are run down at regular intervals and
this procedure provides forcing to the foundation over the frequency range
1059
of operation. By measuring the response at the bearing pedestals (which
is already performed for condition monitoring purposes) an input-output
relation for the foundation may be obtained.
Lees [2] developed a least-squares method to calculate the foundation
parameters by assuming that an accurate model exists for the rotor, that
the state of unbalance is known from balancing runs, and that the dynamic
stiffness matrices of the bearings can be calculated. Although bearing mod¬
els are not in fact well characterised, Lees and Friswell[3] showed that the
parameter estimates are only sensitive to the bearing stiffnesses over limited
frequency ranges, which can be calculated.
Feng and Hahn [4] followed a similar approach but added extra informa¬
tion by measuring the displacements of the shaft. Zanetta[5] also measured
the shaft displacements but included the bearing characteristics as param¬
eters to be estimated. Although any extra information is desirable in a
parameter estimation routine the equipment necessary to measure these
quantities only exists in the newer power stations, and it was desired to
make the method applicable to older plant as well. In the analysis presented
here, the measured data consists of the motion of the bearing pedestals in
the horizontal and vertical directions, although not necessarily in both di¬
rections at every bearing.
2 THEORY
2.1 Force estimation
If D is the dynamic stiffness matrix of a structure defined as
D{u) K ^luC -oj^M (1)
where M,C,K are the mass, damping and stiffness matrices then
Dx = f (2)
where x is response and f force. Referring to figure 1 it is seen that the rotor
is connected to the foundation via the bearings. It is assumed that good
models exist for both rotor and bearings, and that the state of unbalance
is known. The implications of these assumptions will be discussed later.
The dynamic stiffness equation for the whole system may be written as
The subscripts r and / refer to the rotor and foundation degrees-of-freedom
respectively, u refers to the unbalance forces and b to the bearing forces.
1060
ROTOR
Figure 1: Rotor-bearing system
There is a negative sign before the bearing forces ft, since they refer to the
forces acting on the bearings. The foundation d.o.f are those where the re¬
sponses are measured, in other words no internal d.o.f are represented. Df
therefore represents a reduced dynamic stiffness matrix. The response mea¬
surements will not be the total vibration level at the bearings but rather the
vibrations at once-per-revolution and it is assumed that no dynamic forces
at this frequency will be transmitted to the foundation via the substructure
onto which it is fixed.
The equation for the bearings in the global coordinate system is
/ Dbrr Dbrf W ^7- ^ ^ fbT\
V Dij, Dill )\^s) \fbs)
(4)
This assumes that the bearings behave as complex springs, in other
words they have negligible inertia and no internal d.o.f. Substituting (4)
into (3) we have
/ Dt„ Dtrl \ ( ^'■ \ = (
V A/r A// J [ J A ° /
where
/ Dtrl 0 \ f Dirr Di^l \\
[Dtlr DtllJ-[[ 0 DlJ + [Diir Dill))
(5)
(6)
1061
and where subscript t refers to the total model. Solving equation (5) for
Xr leads to
Xr — J^trr ifur ~~
and solving equation (4) for ft / yields
fbf — ^fhr ~ ~^brr^r ~ ^brf^f (S)
All quantities in equation (8) are known either from assumed models {Dr, D^)
or experiment {xf, /^r)- This calculated force fbf may then be used to¬
gether with the measured responses to estimate the foundation parameters.
2.2 Foundation parameter estimation
Once the forces have been estimated, the foundation parameters must
be derived. The dynamic stiffness equation for the foundation is
DfXf — fbf (9)
Although D/ is a reduced stiffness matrix it is assumed that it has the
form of equation (1). Therefore equation (9) may be written as
W{u)v = fbf (10)
where is a column vector formed from the elements of M, C and K and
W is a matrix formed from the response vector which depends explicitly on
u. Clearly this is an under-determined set of equations, but by taking mea¬
surements at many frequencies it may be made over-determined, and thus
solvable in a least squares sense. Since the magnitude of the mass, damp¬
ing and stiffness elements normally differ by several orders-of-magnitude, it
was found expedient to scale the mass parameters by o;^, and the damping
parameters by uJ, where u is the mean value of the frequency.
2.3 Errors in estimates
It is necessary now to examine the effect of errors on the parameter esti¬
mates. Equation (10) is of the form Ax = b, where A has dimension mxn.
In this particular case, A depends on the measured response Xf, whilst b
depends on the measured response, applied unbalance, and assumed rotor
and bearing models. Therefore the estimated parameters will be sensitive
to the following errors:
1. Errors in the rotor model
2. Errors in the bearing model
3. Errors in the state of unbalance
1062
4. Errors in the measured foundation response
The rotor model is generally well known, as is the state of unbalance,
so the main source of error in the estimates is due to measurement noise
and bearing uncertainty. If the least squares problem is formulated as
Ax = b (11)
then
(Aq + Ajv)ic = bo + b/i/ (12)
where the subscript N refers to noise and 0 to data which is noise-free. The
least-squares estimate is given by
X = (Aq Ao + A^Aiv + + A^Ao) ^(Aq + Ayv)^(bo + b;v) (13)
Even if the noise on the outputs is uncorrelated with the noise on the inputs
the expected value of x does not equal that of its estimate:
E[x] - E[x] 0 (14)
In other words the estimate x is biased [6]. In order to reduce the bias of
the estimates, the instrumental variables method can be used. Essentially,
it requires the use of a matrix that is uncorrelated with the noise on the
outputs, but which is strongly correlated with the noise-free measurements
themselves. If W is the instrumental variable matrix, then
W'^Ax = W'^b (15)
Expanding
i = (W^Ao + + b^) (16)
This means that E[x] = x, in other words unbiased estimates result.
Fritzen[7] suggested an iterative method for solving for the parameters.
Initially, equation (10) is solved in a least-squares sense, and the values
of the estimated parameters are used to calculate outputs for the model.
These outputs are then used to create W in the same way as the original
outputs were used to create A, new estimates are obtained, and if neces¬
sary the process is repeated. Experience seems to suggest good convergence
properties[7].
3 SIMULATION
The method under discussion in this paper was tested on a model of a
small test rig located at Aston University, Birmingham. This consists of
a steel shaft approximately 1.1m long, with nominal diameter 38mm. The
shaft is supported at either end by a journal bearing of diameter 100mm,
1063
1 2 3 4 5 6 7 8 9 10 11 12
Figure 2: Rotor-bearing system
L/D ratio of 0.3 and clearance of 25/.tm. There are two shrink-fitted bal¬
ancing discs for balancing runs. Each bearing is supported on a flexible
pedestal to simulate the flexible foundations encountered in power station
turbines. At present these pedestals are bolted onto a massive lathe bed.
The rotor is powered by a DC motor attached via a belt to a driving pulley,
which is in turn attached via a flexible coupling to the main rotor shaft.
A schematic of the rig is shown in figure 2. Dimensions of each station
and material properties are given in table 1. A finite element model of the
rotor with 23 elements was created and short bearing theory was used to
obtain values for the bearing stiffness and damping[8].
The pedestals themselves consist of two rectangular steel plates, 600mm
X 150mm which have two channels cut into them, and which are supported
on knife-edges (figure 3). The vertical stiffness arises from the hinge effect
of the channels, whilst the horizontal stiffness is as a result of the shaft
centre tilting under an applied load. Treating the supports as beams, the
theoretical stiffnesses are:
Ky = 0.^5MN/m !<:, = 1.5MiV/m
where x and y refer to the horizontal and vertical directions respectively.
The masses and damping factors were taken as:
il/4 = My = 50/cp Ca: = Cy = 150iV • s/m
The estimation theory was tested using this model. The finite element
model was used to generate responses at the bearings for frequencies from
1064
Table 1: Table of rotor rig properties
Shaft Properties
Station
Length (mm)
Diameter (mm)
E (GPa)
P (kg/m^)
1
6.35
38.1
200
7850
2
25.4
77.57
200
7850
3
50.8
38.1
200
7850
4
203.2
100
200
7850
5
177.8
38.1
200
7850
6
50.8
116.8
200
7850
7
76.2
38.1
200
7850
8
76.2
109.7
200
7850
9
76.2
38.1
200
7850
10
50.8
102.9
200
7850
11
177.8
38.1
200
7850
12
203.2
100
200
7850
Balancing discs
Station
Length (mm)
Diameter (mm) Unbalance (kg • m)
6
25.4
203.2
0.001
25.4
203.2
0.001
0 to 30 Hz with a spacing of 0.1 Hz. The responses were corrupted by
normally distributed random noise with zero mean and standard deviation
of 0.1% of the maximum response amplitude (applied to both real and
imaginary parts of the response). At each frequency the bearing static
forces were disturbed by noise drawn from a uniform distribution spanning
an interval of 20% of the force magnitude, to introduce uncertainty into the
bearing parameters. The unbalance was assumed to be exactly known. A
series of 30 runs was performed, foundation parameter estimates calculated
and the mean and standard deviations of these estimates obtained.
The magnitudes of the responses at both bearings are given in figure 4,
which show that there are four critical speeds in the frequency range under
consideration. A sample of true and estimated forces in the bearings are
shown in figures 5 and 6.
The means {fi) and standard deviations (a) of the least-square (LS)
and instrumental variable (IV) estimates for the foundation parameters
are shown in table 2.
Displacement (m)
Figure 3: Flexible bearing supports
Figure 4: Magnitudes of responses at bearings
1066
Table 2; Parameter estimates for foundations showing uncertainty
1069
4 DISCUSSION
The results in table 2 show a clear improvement in parameter estimates
when the instrumental variable method is used. There is a clear bias in
the least-squares estimates which is significantly less when the instrumen¬
tal variable method is employed. Also, despite the fact that the bearing
parameters are assumed to be seriously in error, the estimates appear to be
insensitive to them. This will be true provided that the bearings are much
stiffer than the foundation (a reasonable assumption in practice). It does
appear however that in some cases the standard deviation of the instru¬
mental variable estimate is larger than that of the least-squares estimate,
a fact which warrants further investigation.
As far as the rotor model is concerned, impact tests, which are per¬
formed on rotors prior to them entering service, normally give experimental
frequencies which are within a few percent of the theoretical ones. Thus
the assumption that the rotor model is accurately known would appear to
be reasonable.
The state of unbalance may in theory be established from a balancing
run. If two successive run-downs are performed, one due to the unknown
system unbalance and one with known balance weights attached, then pro¬
vided the system is linear the response measurements may be vectorially
subtracted to give the response due to the known balance weights alone.
In order to ascertain the effect of unbalance uncertainty on the parameter
estimates, one run was performed assuming no error in the unbalance. It
should be noted that this assumes that the system is linear.
5 CONCLUSIONS
A method of estimating turbogenerator foundation parameters from
potentially noisy measurement data is demonstrated. It is shown that
making use of the instrumental variable method reduces the bias in the
estimates and improves them quite significantly.
6 ACKNOWLEDGEMENT
The authors wish to acknowledge the support and funding of Nuclear
Electric Ltd and Magnox Electric Pic. Dr Friswell wishes to acknowledge
the support of the Engineering and Physical Sciences Research Council
through the award of an advanced fellowship.
1070
REFERENCES
[1] A. W. Lees and I. C. Simpson. The dynamics of turbo-alternator foun¬
dations: Paper C6/83. In Conference on steam and gas turbine founda¬
tions and shaft alignment, Bury St Edmunds, 1983, IMechE, pp37-44.
[2] A. W. Lees. The least squares method applied to identify ro¬
tor/foundation parameters: Paper C306/88. In Proceedings of the Inter¬
national Conference on Vibrations in Rotating Machinery, Edinburgh,
1988, IMechE, pp209-216.
[3] M. 1. Friswell and A. W. Lees. Estimation of forces exerted on machine
foundations. In M. 1. Friswell and J. E. Mottershead, editors, Interna¬
tional Conference on Identification in Engineering Systems, Swansea,
1996, pp793-803.
[4] N. S. Feng and E. J. Hahn. Including foundation effects on the vibra¬
tion behaviour of rotating machinery. Mechanical Systems and Signal
Processing, 1995, 9, pp243-256.
[5] G. A. Zanetta. Identification methods in the dynamics of turbogener¬
ator rotors: Paper C432/092. In IMechE Conference on Vibrations in
rotating machinery, Bath, 1992. IMechE, ppl73-181.
[6] J. Schoukens and R. Pintelon. Identification of linear systems. Perga-
mon Press, 1991.
[7] C. P. Fritzen. Identification of mass, damping and stiffness matrices of
mechanical systems. Journal of Vibration, Acoustics, Stress and Relia¬
bility if Design, 1986 108, pp9-17.
[8] D. M. Smith. Journal bearings in Turbomachinery. Chapman and Hail,
1969.
1071
1072
SHELL MODE NOISE IN RECIPROCATING REFRIGERATION
COMPRESSORS
Ciineyt Oztiirk and Aydin Bahadir
Tiirk Elektrik Endiistrisi A.$
R&D Department
Davutpa§a, Litres Yolu, Topkapi -34020, Istanbul, Turkey
ABSTRACT
This study describes the successful endeavor to understand the causes of noise
that appear on the shell modes of the reciprocating refrigeration compressors.
The compressor shell is generally considered as the acoustic enclosure that
reflects the acoustic energy back into the compressor cavity but also as the
transmitter and radiator of the transmitted acoustic energy that could be
radiated into the air or transmitted to the structure. Vibrations of the
compressor shell can easily be characterized in terms of the modal parameters
that consist of the natural frequencies, mode shapes and damping coefficients.
The noise source harmonics and the shell resonances couple to produce the
shell noise and vibration. The harmonic spacing is equal to the basic pumping
frequency. Results of the studies indicate that important natural frequencies of
the compressor shell usually stay between 2000-6000 Hz interval. The
important natural frequencies are first natural frequencies in the lower range
with the longer wavelengths that radiate well.
INTRODUCTION
Compressor noise sources are those processes where certain portions are
separated from the desired energy flow and transmitted through the internal
components of compressor to the hermetic shell where it is radiated from the
shell as airborne noise on vibration of supporting structure will eventually
radiate noise from some portion of the structure. Noise sources of the
reciprocating refrigeration compressors can be classified as motor noise,
compression process noise and valve port flow noise.
1073
In reciprocating compressors there is very high density of noise harmonics even
though they decay in amplitude at high frequencies. Generally, these noise
source harmonics and the shell resonances couple to produce shell noise and
vibration
NOISE GENERATION MECHANISM OF THE RECIPROCATING
REFRIGERATION COMPRESSORS
Significance of the problem
The results of the sound radiation characteristics shown at figure 1 indicated
that certain high amplitude frequency components had very distinctive sound
radiation patterns. It was suspected these frequencies correspond to excitation
of either structural resonances of the compressor shell or acoustic resonances of
the interior cavity space. Resonances those amplify the noise and vibration
caused by pumping harmonics of a compressor and thus can be the cause of
significant noise problems.
Sound power - A-Wcighted
— I - , - 1 - , - ; - P — — r — f — ! — — 'I' "I '"'I — I - ”r ' i i
50.0 63.0 80,0 100,0 125,0 160,0 200,0 250,0 315,0 400,0 500,0 630,0 800,0 l,0k t,3k 1,6k 2,0k 2,5k 4,0k S,0k 63k S,0k 10,0k
Frequency [Hz]
Figure 1, Noise Radiation Characteristics of Reciprocating Refrigeration
Compressor.
Noise sources
Noise in a compressor is generated during cyclic compression, discharge,
expansion and suction process. The character of noise sources is harmonic due
to periodic nature of the compression process. These harmonics are present in
the compression chamber, pressure time history and loading of the compressor
through drive system. The motor can not provide immediate response to load
harmonics and load balance is obtained at the expense of acceleration and
1074
deceleration of the motor drive system. Harmonic vibrations of the motor drive
system can then excite the resonant response in the compressor components
that can transmit the acoustic energy in very efficient way. The rest of the noise
sources are, turbulent nature of flow depending to passage through valve ports,
valve impacts on their seats and possible amplification when matched with
mechanical resonances.
MECHANICAL FEATURES OF SHELL
Mechanical features of the compressor
The hermetically sealed motor compressor comprises in general a motor
compressor unit including a motor assembly mounted with a frame and a sealed
housing within which the compressor is supported by means of plurality of coil
springs each having one end spring with the frame and the other end connected
with the interior of the housing.
The refrigerant gas as it is compressed in the cylinder is discharged through the
discharge chamber in the cylinder head into the discharge muffler. The
discharge muffler is generally mounted on the cylinder head attached in
covering relation to an end face of the cylinder. Where the sealed casing is
spherical in shape for better noise suppression, an upper end of the cylinder
head tends to interfere with an inner wall surface of the casing, a disadvantage
that can only be eliminated by increasing the size of scaled casing for providing
a desired hermetically sealed motor.
Compressor Shell
The shell is easily be characterized with the modal parameters. The ideal shell
should be designed in a way that keep all the excitation frequencies at the mass
controlled region of all its modes. But, depending on the very tight constraints
that come with the gas dynamics and motor locations, it is not allowed to be
flexible during the design of shell. As a consequence of the existing design
limitations natural frequencies of the shell usually fall between 1000-5000 Hz.
SHELL RADIATION
Figure 1 illustrates how the sound pressure level of a pumping harmonic can
increase as it nears a resonant frequency. The sound pressure level of the
pumping harmonic increases around the shell resonances. The resonance
generally radiates primarily from the large flat sides of the compressor. There
are three major acoustic cavity resonances 400, 500 and 630 Hz and four major
structural resonances of the compressor shell: around 1.6 K, 2 K, 2.5 and 3.2
K. Hz at which noise radiates well in certain directions.
1075
SHELL RESONANCES
Figures 2, 3 and 4 illustrate the frequency responses of the compressor shell
when measured with the impact hammer method. Figure 2 is the response to
the excitation in x direction, figure 3 is for the excitation direction and figure 4
is for the excitation in z direction.
Figure 2, Frequency response of the compressor shell when excited in X
direction.
Figure 3, Frequency response of the compressor shell when excited in Y
direction.
1076
Figure 4, Frequency response of the compressor shell when excited in Z
direction.
To verify the hypothesis that resonances were contributing to some of the noise
problems of the reciprocating piston compressor, a modal analysis of the shell
and interior cavity was performed.
For the shell modal analysis, the accelerometer to measure the response
remained stationary, while the impact location was moved. The test was
performed in this manner for convenience since it was easier to fix the
accelerometer in one location and strike the compressor with force hammer at
each grid point to obtain transfer function for each measurement location.
Identical results are obtained if impact occurred at a single point and the
response was measured at each measurement location. Preliminary test were
performed initially to identify an appropriate measurement location at which all
important natural frequencies of shell are detectable. Several force input and
response locations were evaluated to determine the best location to mount
accelerometer to measure the shell response.
The shell resonances are also calculated by using the Structural FEM analysis.
Table- 1 lists the natural frequencies predicted in these studies. During the finite
element analysis, the models of the compressor were built, based on the CAD
models. The shell consists of 7500 elements. The mesh densities are quite
adequate for the structural analysis in the frequency range of interest. In order
to investigate the possible influence of the crank mechanism on the natural
frequencies of the shell, a simple model of the crank mechanism was introduced
to the FE model. During the calculations, the crank mechanism was simplified
as a rigid block with certain mass and rotary inertia and modeled with solid
elements. The shell and the crank mechanism have been suspended with the
1077
springs from 3 positions and in all 6 transitional and rotational directions.
Depending on the negligible spring effect on the longitudinal direction,
estimated values have been used in 5 directions. The FE model has been
assumed to be free-free.
Mode
#
Frequency
Hz
Mode
#
Frequency
Hz
1
1997
21
4716
2
22
4925
3
2293
23
4994
4
24
5
25
5159
6
26
5454
7
2889
27
5476
8
3258
28
9
29
EESI^HI
10
30
5783
11
3376
31
5936
12
3551
32
5999
13
3577
33
6035
14
34
6055
15
3788
35
6183
16
3958
36
6237
17
4383
37
6314
18
4481
38
19
4644
39
20
4702
40
6701 1
Table- 1, Calculated Natural Frequencies of the Shell
Figures 5, 6, 7 and 8 illustrate how the mode shapes of the shell vary at the
mode frequencies of 2754, 3332, 3551 and 3788 Hz , These figures indicate
that the shell vibrates predominantly along the large flat sides of the compressor
at points where the suspension springs are attached to the shell wall at these
natural frequencies. When referred to figure 1 of the noise radiation this
frequency range is also the range where the noise radiates efficiently from the
large flat sides of the compressor. Thus, there is good correlation between the
acoustic data and structural data for these frequencies. The slight discrepancies
in the structural natural frequencies and the acoustic data. Acoustic data have
been picked up at the shell temperature of the reciprocating piston compressor
that could reach up to 1 10 C.
The modal analysis results also indicate that the compressor suspension springs
are attached to a point on the shell where the shell is comparatively compliant.
Thus, the vibrational energy transmitted through the springs to the compressor
shell can and did effectively excite the shell vibrations. Also, significant shell
vibrations occur along the large flat sides of the compressor shell indicating the
curvature of the shell needs to be increased to add stifihiess to the shell.
Based on the results of the shell modal analysis, it is recommended the
suspension springs moved away from the compliant side walls of the shell. A
four spring arrangement at the bottom of the shell near corners where the
curvature is sharp would reduce the amount of vibration energy transferred to
the shell because of the reduced input mobility of the shell at these locations.
It is also believed increasing the stiffness of the shell by increasing the curvature
will provide noise reduction benefits. The greater shell stiffness lowers the
amplitude of the shell vibrations. Figure 9, illustrate the third octave change in
compressor noise with the same compressor in the new shell. An over all noise
level of 5 dBA has been obtained.
Figure 9 Compressor noise level improvement after the shell modification.
The increased shell stiffness also raises the natural frequencies of the shell
where there is less energy for transfer function response. However, there is a
possible disadvantage to increasing stiffness of the shell. The higher natural
frequency lowers critical frequency of the shell thus reducing transmission loss
of the shell.
1081
Damping treatments can also have obvious benefits in vibration and noise
reduction. Visco elastic and Acoustic dampings are considered to avoid the
shell excitations. The application of dampers can also provide up to 5 dBA
reductions when appropriately located on the shell.
CONCLUSION
The results of studies indicate that structural resonances of the shell are indeed
amplifying the noise due to the pumping harmonics of the reciprocating
refrigeration compressor to cause significant noise radiation outside of
compressor.
In order to tackle with this noise problem, within the scope of these studies
two different effective shell noise control are considered based on the results of
numerical and experimental structural analysis and acoustic features of the
reciprocating compressor. First, shell noise control method is the redesign of
the shell with increased stiffness by replacing all the abrupt changes in the
curvature with the smooth continuous changes. It is apparent that change in the
shell configuration can shift the first shell resonance from 1750 Hz to nearly
3200 Hz. The results of the redesign effects can reach up to 3-5 dBA reduction
on third octave noise levels. Second treatment that could be applied against the
excitation of shell resonances are considered as the acoustic and viscoelastic
dampers. These dampers can be chosen to operate efficiently at the shell
resonant frequencies. These two applications can also provide up to 2-5 dBA
reduction on the third octave band of the noise emission but the long term
endurance and temperature dependence of these materials can always be a
question mark when considered from the manufacturer point of view.
REFERENCES
1. JFROBATTAand ID. JONES 1991, Purdue University, School of
Mechanical Engineering, The Ray Herrick Laboratories, Report no : 1912-1
HL 91-9P, 73-84, Investigation of Noise Generation Mechanisms and
Transmission Paths of Fractional Horsepower Reciprocating Piston and
Rolling Piston Compressors
2. HAMILTON I F 1988, Purdue University , School of Mechanical
Engineering, The Ray Herrick Laboratories, 207-213 Measurement and
Control of Compressor Noise
3. C OZTURK, A AQIKGOZ and J L MIGEOT 1996, International
Compressor Engineering Conference at Purdue, Conference Proeceeding ,
Volume II, 697-703, Radiation Analysis of the Reciprocating Refrigeration
Compressor Casing
1082
A COMPARATIVE STUDY OF MOVING FORCE IDENTIFICATION
T.H.T. Chan, S.S. Law, T.H. Yung
Department of Civil & Structural Engineering,
The Hong Kong Polytechnic University, Hung Horn, Kowloon, Hong Kong
ABSTRACT
Traditional ways to acquire truck axle and gross weight information are
expensive and subject to bias, and this has led to the development of Weigh-
in-Motion (WIM) techniques. Most of the existing WIM systems have been
developed to measure only the static axle loads. However dynamic axle loads
are also important. Some systems use instrumented vehicles to measure
dynamic axle loads, but are subject to bias. These all prompt the need to
develop a system to measure the dynamic axle loads using an unbiased
random sample of vehicles. This paper aims to introduce four methods in
determining such dynamic forces from bridge responses. The four methods are
compared with one another based on maximum number of forces to be
identified, minimum number of sensors, sensitivity towards noise and the
computation time. It is concluded that acceptable estimates could be obtained
by all the four methods. Further work includes merging the four methods into
a Moving Force Identification System (MFIS).
INTRODUCTION
The truck axle and gross weight information have application in areas
such as the structural and maintenance requirements of bridges and pavements.
However, the traditional ways to acquire that are expensive and subject to bias,
and this has led to the development of Weigh-in-Motion (WIM) techniques.
Some systems are road-surface systems which make use of piezo-electric
(pressure electricity) or capacitive properties to develop a plastic mat or
capacitive sensors to measure axle weight [1]. Another kind of WIM system is
the under-structure systems in which sensors are installed under a bridge or a
culvert and the axle loads are computed from the measured responses e.g.
AXWAY [2] and CULWAY [3]. All the above mentioned systems can only
give the equivalent static axle loads. However dynamic axle loads are also
important as they may increase road surface damage by a factor of 2 to 4 over
that caused by static loads [4]. Some systems use instrumented vehicles to
measure dynamic axle loads [5], but are subject to bias. These all prompt the
need to develop a system to measure the dynamic axle loads using unbiased
random samples of vehicles. Four methods are developed to determine such
1083
dynamic forces from bridge responses which include bending moments or
accelerations.
EQUATION OF MOTIONS FOR MOVING LOADS
The moving force identification methods described in this paper are the
inverse problems of an predictive analysis which is defined by 0‘ Connor and
Chan [6] as an analysis to simulate the structural response caused by a set of
time-varying forces running across a bridge. Two models can be used for this
kind of analysis.
A. Beam-Elements Model
0‘ Connor and Chan [6] model the bridge as an assembly of lumped
masses interconnected by massless elastic beam elements as shown in Figure 1,
and the nodal responses for displacement or bending moments at any instant
are given by Equations (1) and (2) respectively.
Moving Loads
... ip. Ip, ^
- D — D — 2^
Lumped Masses 1 2 ... N-1 N
Figure 1 - Beam-Elements Model
{Y} = [Y,]{F}-lY,][Am]{n-m[C]{Y} (1)
{5M} = [5MJ{P}-[5M;][Am]{y}-[5MJ[C]{f} (2)
where [P] is a vector of wheel loads, [Am] is a diagonal matrix containing
values of lumped mass, [C] is the damping matrix, 5M,L,7,7are the nodal
bending moments, displacements, velocities and accelerations respectively,
[R,^] {'R ’ can be Y or BAP) is an m x n matrix with the ith column representing
the nodal responses caused by a unit load acting at the position of the ith wheel
load and [i?J {'R' can be Y or BM) is an n x n matrix with the ith column
representing the nodal responses caused by a unit load acting at the position of
ith internal node.
1084
B. Continuous Beam Model
Assuming the beam is of constant cross-section with constant mass per
unit length, having linear, viscous proportional damping and with small
deflections, neglecting the effects of shear deformation and rotary inertia
(Bernoulli-Euler’s beam), and the force is moving from left to right at a
constant speed c, as shown in Figure 2, then the equation of motion can be
written as
P
d-v{xj)
a-
a
+ EI
^v(x,0
a-
S(x-ct)f{t)
(3)
where v(x,0 is the beam deflection at point x and time t; p is mass per unit
length; C is viscous damping parameter; £ is Young's modulus of material; I
is the second moment of inertia of the beam cross-section; f(t) is the time-
varying force moving at a constant speed of c, and Sft) is Dirac delta function.
\/(0
— O
Figure 2 - Simply supported beam subjected to a moving force f(t)
Based on modal superposition, the dynamic deflection y(x,t) can be
described as follows;
v{x,t) = Y,<i'„{x)q„{t)
}}=\
(4)
where n is the mode number; 0n(x^) is the mode shape function of the n-th
mode and qyi(t) is the n-th modal amplitudes.
Based on the above mentioned predictive analyses, four Moving Force
Identification Methods (MFIM) are developed.
1085
FIRST MOVING FORCE IDENTIFICATION METHOD
INTERPRETIVE METHOD I - BEAM-ELEMENTS MODEL (IMI)
It is an inverse problem of the predictive analysis using beam-elements
model From Equation (1), it can be seen that if Y is known at all times for all
interior nodes, then 7 and Y can be obtained by numerical differentiation.
Equation (1) becomes an overdetermined set of linear simultaneous equations
in which the P may be solved for them. However a particular difficulty arises
if measured BM are used as input data. Remembering that the moving loads P
are not normally at the nodes, the relation between nodal displacements and
nodal bending moments is
{r}^[Y,]{BM} + [Yc]{B} (5)
where allows for the deflections due to the additional triangular
bending moment diagrams that occur within elements carrying one or more
point loads P. [}^]can be calculated from the known locations of the loads.
[}^^] and {BM}aic known, but {7} cannot be determined without a
knowledge of (P}. 0‘Connor and Chan [6] describe a solution uses values of
{P} assumed from the previous time steps.
SECOND MOVING FORCE IDENTIFICATION METHOD
INTERPRETIVE METHOD II - CONTINUOUS BEAM MODEL (IMII)
From the predictive analysis using continuous beam model, if the ith-
iTTX
mode shape function of the simply supported Euler‘s beam is sin-—-, then
the solution of Equation (3) takes the form
v = ^sin— F;(0 (6)
/=i L
where V.(t) , (i = 1, 2, • • •) are the modal displacements.
Substitute Equation (6) into Equation (3), and multiply each term of
Equation (3) by the mode shape function sin(y;r.x / L) , and then integrate the
resultant equation with respect to x between 0 and L and use the boundary
conditions and the properties of Dirac function. Consequently, the following
equation can be obtained
1086
op
u 2 El ^
where = —5 — - C(y, =
C
2/rryy,
at the j-th mode.
If there are more than one moving loads on the beam, Equation (7) can be
written as
. 7r(ct-Xk)
._7r(ct~x,)
7r{ct-x^) ■
■^.1
'2CM'
'cojv;
sin - ^
L
sin - ^ •
L
sin - —
L
'K
V,
2C,co,K
+
CO IVj
_ 2
. 27r(ct-Xt)
sin
. 27r(ct~x^}
sm - =- ■
. 27r{ct - x. )
•• sm — - —
Pi
~ Ml
A
.1
_2C,.a)/„_
_coX_
. n7r(ct-x,)
sin — ^ ^
. riKict-x^)
sin — ^ - — .
. n7c{ct-x^)
•• sin — - —
A.
L
L
L
(8)
in which is the distance between the k-th load and the first load and x, = 0 .
If Pi Pk are constants, the closed form solution of Equation (3) is
/ N ^ 1 • ■ j7E{ct-Xi) a . . . , ^ {Q\
48E/f:i' L \ L j ^ J
nc
in which a =
Lcd
If we know the displacements of the beam at x, , x, , • • • , , the moving
loads on the beam are given by
{v} = [S, „.]{/>} (10)
in which {v}=[v| v, ■■■ v,]' {•^’} = [-^1 ^2 ■"
^n-
, where
L" A 1 . j7rx,J . J7r{ct-x.) a . , . , ,
= 7577X^2772 - 27^“ “7~ - 7 - /c)
48£:/^;-0- -a ) L \ L j
1087
If l>k, that means the number of nodal displacements is larger than or
equal to the number of axle loads, then according to the least squares method,
the equivalent static axle load can be given by
{/>}=([s,,]''[s„])''[s,q^{v} (11)
If the loads are not constant with time, then central difference is used to
proceed from modal displacements to modal velocities and accelerations.
Equation (8) becomes a set of linear equations in which P^. for any instant can
be solved by least squares method. Similar sets of equations could be obtained
for using bending moments to identify the moving loads.
THIRD MOVING FORCE IDENTIFICATION METHOD
TIME DOMAIN METHOD (TDM)
This method is based on the system identification theory [7]. Substituting
Equation (4) into Equation (3), and multiplying each term by 0j(x), integrating
with respect to x between 0 and L, and applying the orthogonality conditions,
then
dt-
dt
M.,
(12)
where con is the modal frequency of the n-th mode; is the damping ratio of
the n-th mode; Mn is the modal mass of the n-th modt, pn(t) is the modal force
and the mode shape function can be assumed as 0„(x) = sin(^;w / L) .
Equation (12) can be solved in the time domain by the convolution
integral, and yields
= (13)
^11 0
where ~ sin(^u„0, t >0 (14)
and co\, (15)
Substituting Equation (13) into Equation (4), the dynamic deflection of
the beam at point x and time t can be found as
v(x,0 = y — ^sin^^ sin6>|,(^ - '^)sin^^^^/(T)dr (16)
fxpLo),, L i L
1088
A. Force Identification from Bending Moments
The bending moment of the beam at point x and time t is
m{xj) = -El -
3c-
(17)
Substituting Equation (16) into Equation (17), and assuming the force f(t)
is a step function in a small time interval and f(t) =0 at the entry and exit, then
let
^ 2El7r' n~ . yitu: ^
C,„ = — 73 - ^sm— Ar,
pi co„ L
{k) = sin(<:y'„ A/A:),
^ . ,n7(cNi
52(*) = sm(— ^/c)
(18)
(19)
N„=-
Equation (17) can be expressed as
cEt
m(2)]
0
... 0 '
/(i)
m(3)
<
■=S c,„
/) = 1
E;,S,{2)S,i\)
e:S,{\)S,{2)
... 0
/(2)
m{N)
_E::"S,iN-\)S,il)
E;^--S,{N~2)S,i2)
Kc.
(20)
where A^ is the sample interval and N+I is the number of sample points, and
f>„=£r'''“*'S,(Al-A^„+l)S,(//„-l)
Equation (20) can be simplified as
B f = m (21)
(A'-l)x(;V/^-|) ('V/^->)xl (,V_l)xl
U N = matrix B is a lower triangular matrix. We can directly find
the force vector f by solving Equation (21). If 77 > and/or 77/ bending
moments (77/ > 1) are measured, least squares method can be used to find
the force vector f from
1089
(22)
B, ■
m,
f
m.
®-v,.
The above procedure is derived for single force identification. Equation
(21) can be modified for two-forces identification using the linear
superposition principle as
“B. 0
B, B,
B. B,
where B 3 [N^x (Nb-1)] , Bt [(N -1-2NJx (Ng-l)], and B^ [NjX (Nb -1)] are
sub-matrices of matrix B. The first row of sub-matrices in the first matrix
describes the state having the first force on beam after its entry. The second
and third rows of sub-matrices describe the states having two-forces on beam
and one force on beam after the exit of the first force.
B. Identification from Bending Moments and Accelerations
Similarly the acceleration response of the beam can be expressed as
A f = V
(24)
The force can also be found from the measured acceleration from
Equation (24). If the bending moments and accelerations responses are
measured at the same time, both of them can be used together to identify the
moving force. The vector m in Equation (21) and v in Equation (24) should be
scaled to have dimensionless unit, and the two equations are then combined
together to give
A/iv||J [v/llvlj
(25)
where Ihli is the norm of the vector.
FOURTH MOVING FORCE IDENTIFICATION METHOD
FREQUENCY DOMAIN METHOD (FDM)
Equation (12) can also be solved in the Frequency Domain. Performing
the Fourier Transform for Equation (12),
1090
1
1
(26)
where
co;, -CO- +24„co„co M„
W
i
—00
(27)
(28)
Let
_ 1 _
co;,-co- +2^„a„a
(29)
Hn(co) is the frequency response function of the n-th mode. Performing the
Fourier Transform of Equation (4), and substituting Equations (26) and (29)
into the resultant equation, the Fourier Transform of the dynamic deflection
v(x,t) is obtained as
A. Force Identification from Accelerations
Based on Equation (30), the Fourier Transform of the acceleration of the
beam at point .x and time t can be written as
V(x,a) = -®^X^(D„(x)//„(cy)P„(cy) (31)
Considering the periodic property of the Discrete Fourier Transform (DFT),
and let
A/^m-
Equation (32) can be rewritten as
(32)
1091
,m = 0,l, (33)
V{m) = X H„(m)'i'„(m)[F,{0) + iF,(0)]
/;=1
A^/2-1 « _
k=\ H=t
Nn~\ =0 _
/t = l /) = l
+E /2)[F„(A^ /2) - iivCA^ / 2)]
«=I
where is the Fourier Transform of the n-th mode shape, and F is the
Fourier Transform of the moving force.
Writing Equation (33) into matrix form and dividing F and V into real and
imaginary parts, it yields
Because F, (0) = 0, Fj (N / 2) = 0,Vj (0) = 0,F; ( / 2) = 0 , Equation (34) can
be condensed into a set of N order simultaneously equations as
(35)
Fr and F, can be found from Equation (35) by solving the Nth order linear
equation. The time history of the moving force f(t) can then be obtained by
performing the inverse Fourier Transformation.
If the DFTs are expressed in matrix form, the Fourier Transform of the
force vector f can be written as follows if the terms in f are real [8].
F = — Wf (36)
N
whereW = e''^‘‘^^^ (37)
1092
0 0 0 0 0
0 1 2 ••• -2 -1
0 2 4 ••• -4 -2
0 -2 -4 ••• 4 2
0 -1 -2 2 1
yv X
The matrix W is an unitary matrix, which means
w-‘ =(w* y
(38)
where W* is a conjugate of W. Substituting Equation (36) into Equation (35),
V = — Al
N
W 0
- 1
X
_ Ij
(39)
or
V=^ A fe
(40)
linking the Fourier Transform of acceleration V with the force vector fg of
the moving forces in the time domain. Wg is the sub-matrix of W. If N =
fg can be found by solving the Mh order linear equations. If N> or more
than one accelerations are measured, the least squares method can be used to
find the time history of the moving force f(t).
Equation (40) can be rewritten as follows
(41)
relating the accelerations and force vectors in the time domain. Also if N -
Nb, fe can be found by solving the Nth order linear equation. If N > Nb or
more than one acceleration are measured, the least squares method can be
used to find the time history of the moving force f(t).
If only Nc {Nc ^ N) response data points of the beam are used, the
equations for these data points in Equation (41) are extracted, and described as
V, =(w*)" A
C V B / A/x/V
c
B
Nr X N
W3
/Vx;V,
yV„x!
(42)
1093
In usual cases Nc > Ng, so the least squares method is used to find the time
history of the moving force f(t). More than one acceleration measurements at
different locations can be used together to identify a single moving force for
higher accuracy.
B. Identification from Bending Moments and Accelerations
Similarly, the relationships between bending moment m (and M ) and
the moving force f can be described as follows,
M = ^ B W is (43)
/Vxl N N^Nn
(44)
(45)
The force vector fg can be obtained from the above three sets of
equations. Furthermore, these equations can be combined with Equations (40),
(41) and (42) to construct overdetermined equations before the equations are
scaled. Two forces identification are developed using the similar procedure as
that for the Time Domain Method.
COMPARATIVE STUDY
The first moving force identification method is implemented in a
computer program using FORTRAN, while the other three methods are
implemented under the environment of a high performance numerical
computation and visualization software. The predictive analysis using beam-
elements model is used to generate the theoretical bridge responses and the
four moving force identification methods then use these responses to recover
the original dynamic loads. In this study, if at least 80% of the identified
forces at any instant of any load lie within ± 1 0% of the original input force,
the method is considered acceptable. It is found that all the four methods can
give acceptable results.
It is decided to carry out a preliminary comparative study on the four
methods in order to study the merits and limitations of each method so as to
consider the future development of each method and devise a plan to develop a
1094
moving force identification system which can make use of the benefits of all
the four methods.
A. Maximum Number of Forces
This is to examine the maximum number of axle loads that can be
identified by each method. Theoretically, provided that sufficient number of
nodal sensors are installed, IMI and IMII can be used to identify as many loads
as the system allows. Basically, the number of axle loads cannot be larger than
the number of nodal sensors. Regarding TDM and FDM, as the formulation of
the governing equation is derived for two moving forces, the maximum
number of axle loads that can be identified is two.
B. Minimum Number of Sensors
Based on a study of common axle spacings of vehicles currently
operating on Australian roads, and the cases with zero nodal responses,
0‘ Connor and Chan [6] state the relationships of the minimum number of
sensors used for IMI and the span length of a bridge as follows:
Using bending moment, for span length L > 4.8m,
Min. number of nodal moments required = int{
Using displacements, for span length L > 13.8m,
Min. number of nodal displacement required = INj{
and for span length L <13.8m,
Min. number of nodal displacement required
L - 4.8'
1.7
L-UX
3.7
+ 4
+ 6
(46)
(47)
(48)
For IMII, it is found that the number of sensors required are generally less
than that for IMI. Regarding TDM and FDM, the programs are not as flexible
as that for IMI and IMII and it is not easy to change the number of sensors.
Meanwhile the sensors are fixed to be at 1/4, 1/2 and 3/4 of the span.
C. Sensitivity towards Noise
In general, all the four methods can compute the identified forces exactly
the same as those given to the predictive analysis to generate the
corresponding responses. It is decided to add white noise to the calculated
responses to simulate polluted measurements and to check their sensitivity
towards noise. The polluted measurements are generated by the following
1095
equations:
m = n>o.nkui«.«] x N„,
(49)
where Ep is a specified error level; is a standard normal distribution
vector (with zero mean value and unity standard deviation).
Several cases are studied using Ep =1%, 3%, 5% and 10%. It is found
that when using bending moments for IMI and IMII, and if Ep is less than 3%,
acceptable results can be obtained. For noise which is greater than 3%, a
smoothing scheme should be adopted to smooth the simulated data.
Acceptable results cannot be obtained for Ep > 10%. Besides, both IMI and
IMII cannot give acceptable results when using displacements.
Both TDM and FDM cannot give acceptable results when using
displacements only, accelerations only or bending moments only. In general
TDM and FDM are less sensitive to noise when comparing to IMI and IMII.
They can give acceptable results for Ep up to 5 % without any smoothing of
the polluted simulated data.
D. Computation Time
In general, the computer program for IMI only takes few seconds to
identify moving forces. In order to compare the computation time, IMI is
implemented in the same environment as the other three methods. It is found
that IMI and IMII take about 2-3 minutes to give the identified forces for a
case of two axle loads using a 80486 computer. However, under the same
working conditions, TDM and FDM almost take a whole day for any one of
them to identify two moving forces. It is due to the fact that both of them
require to set up an huge parametric matrix.
CONCLUSIONS
Four methods are developed to identify moving time-varying force and
they all can produce acceptable results. From a preliminary comparative study
of the methods, it is found that IMI and IMII have a wider applicability as the
locations of sensors are not fixed and it can identify more than two moving
forces. However, TDM and FDM are less sensitive to noise and require less
number of sensors. It is decided to further improve the four methods and then
a more detailed and systematic comparison can be carried out afterwards. The
possible development of the methods are described as follows.
1096
Both the IMI and IMII are developed to work with one kind of responses,
e.g. either displacements or bending moments. It is suggested to modify the
programs to use mixed input parameter, e.g. use bending moments as well as
accelerations as that for TDM or FDM. Regarding the TDM and FDM, as the
basic formulations are based on two-axle moving forces, so it is necessary to
modify the governing equations for multi-axle. In addition, the computation
time for TDM or FDM under the environment of the high performance
numerical computation and visualization software used is unbearable. It is
expected that the time will be significantly reduced if the methods are
implemented in programs using standard programming languages like
FORTRAN 90 or C. Then the four methods can be combined together and
merged into a Moving Force Identification System (MFIS) so that it can
automatically select the best solution routines for the identification.
ACKNOWLEDGMENT
The present project is funded by the Hong Kong Research Grants Council.
REFERENCES
1. Davis, P. and Sommerville, F., Low-Cost Axle Load Determination,
Proceedings, 13th ARRB & 5th REAAA Combined Conference, 1986,
Part 6, p 142-149.
2. Peters, R.J., AXWAY - a System to Obtain Vehicle Axle Weights,
Proceedings, 12th ARRB Conference, 1984, 12 (2), p 10-18.
3. Peters, R.J., CULWAY - an Unmanned and Undetectable Highway Speed
Vehicle Weighing System, Proceedings, 13th ARRB & 5th REAAA
Combined Conference, 1986, Part 6, p 70-83.
4. Cebon, D. Assessment of the Dynamic Wheel Forces Generated by
Heavy Vehicle Road Vehicles. Symposium on Heavy Vehicle Suspension
Characteristics, ARRB, 1987.
5. Cantineni, R., Dynamic Behaviour of Highway Bridges Under The
Passage of Heavy Vehicles. Swiss Federal Laboratories for Materials
Testing and Research (EMPA) Report No. 220, 1992, 240p.
6. O'Connor, C. and Chan, T.H.T., Dynamic Wheel Loads from Bridge
Strains. Structural Engineering ASCE, 1 14 (STS), 1988, p. 1703- 1723.
7. Briggs, J.C. and Tse, M.K. Impact Force Identification using Extracted
Modal Parameters and Pattern Matching, International Journal of Impact
Engineering, 1992, Vol. 12, p361-372.
8. Bendat, J.S. and Piersol, J.S., Engineering Application of Correlation and
Spectral Analysis. John Wiley & Sons, Inc. Second Edition, 1993.
1097
1098
ESTIMATING THE BEHAVIOUR OF A
NONLINEAR EXPERIMENTAL MULTI DEGREE
OF FREEDOM SYSTEM USING A FORCE
APPROPRIATION APPROACH
P.A. Atkins J.R. Wright
Dynamics and Control Research Group
School of Engineering, Simon Building, University of Manchester,
Oxford Road, Manchester. M13 9PL
ABSTRACT
The identification of nonlinear multi degree of freedom systems involves a
significant number of nonlinear cross coupling terms, whether the identifi¬
cation is carried out in spatial or modal domains. One possible approach
to reducing the order of each identification required is to use a suitable
pattern of forces to drive any mode of interest. For a linear system, the
force pattern required to drive a single mode is derived using a Force Ap¬
propriation method. This paper presents a method for determining the
force pattern necessary to drive a mode of interest of a nonlinear system
into the nonlinear region whilst the response is controlled to remain in pro¬
portion to the linear mode shape. Such an approach then allows the direct
nonlinear modal terms for that mode to be identified using the Restoring
Force method. The method for determining the relevant force patterns is
discussed. The implementation of the method for experimental systems is
considered and experimental results from a two degree of freedom ’bench¬
mark structure’ are presented.
INTRODUCTION
Force Appropriation [1] is used in the analysis of linear systems to de¬
termine the force patterns which will induce single mode behaviour when
applied at the relevant natural frequency. This technique is used in the
aerospace industry during Ground Vibration Tests: each normal mode of
1099
a structure is excited using the derived force pattern and thus identified
in isolation. Current practice, when the presence of nonlinearity is sus¬
pected, is to increase input force levels and monitor the variation of tuned
frequencies. Some information about the type of nonlinearity present may
be found, but no analytical model can be derived. Thus predictions for
behaviour at higher levels of excitation axe not possible.
A number of techniques for identifying nonlinearity, for example the Restor¬
ing Force method [2], have been demonstrated on systems with low num¬
bers of degrees of freedom. Unfortunately, in practice, structures have a
large number of degrees of freedom, often with a high modal density. A
classical Restoring Force approach to the identification of such systems
could involve a prohibitive number of cross coupling terms. The ability to
treat each mode separately, by eliminating the effects of the cross coupling
terms, would thus.be advantageous. Subsequent tests could then evaluate
the cross coupling terms.
For these reasons it would be useful to extend Force Appropriation to the
identification of nonlinear systems. An approach has been developed [6]
that allows an input force pattern to be derived that will result in a non¬
linear response in the linear mode shape of interest. This force pattern is
derived using an optimisation approach. The mode of interest can then be
identified using a single degree of freedom nonlinear identification method.
In this work the Restoring Force method is used to examine the nonlinear
response of a particular linear mode and an application of this approach
to a two degree of freedom experimental system is presented.
THEORY
The theoretical approach is demonstrated for the two degree of freedom
system with spring grounded nonlinearity shown in figure 1. The equations
of motion for this system in physical space are:
m 0
0 m
+
+
(1 + a)c
—ac
'±1 1
—ac
(1 + d)c
\
X2 J
(l -}- Q^k
—ak
fill
—ak
(1 + a)k
l^x\
0
where is the cubic stifEhess coefficient and a is a constant that allows the
frequency spacing of the natural frequencies to be varied. These equations
can be transformed to linear modal space using the transformation:
{i} = [ij>]{u} (2)
where [(j>] is the modal matrix of the underlying linear system and the
vector {u} defines the modal displacements. For this symmetrical system
1100
the modal matrix is
1 1
1 -1
(3)
The equations of motion transformed to linear modal space using the nor¬
malised modal matrix are:
771 0 \ ill
0 771 (112
+
c 0
0 (1 + 2g)c
1^2/
■jb 0
Ull
0 (1 + 2a)k
1^2]
P{Ul-U2y/4:'
-U2)V4
where {p} is the modal input vector. It can be seen from the above equa¬
tions that the cubic nonlinearity couples the modes in linear mod^ space;
in fact there are a significant number of terms for a single nonlinearity.
The proposed method aims to determine the force pattern that will reduce
the response of this system to that of a single mode.
It was shown in a previous paper [3] that this can be achieved by seeking a
force vector that will cause motion only in the target mode, by eliminating
motion in the coupled mode. In practice, physical data from transducers
are available. Any subsequent transformations would be time consuming.
It is shown below that causing motion in one mode to be zero is equiva¬
lent to forcing motion in a linear mode shape, mode one in this example.
Consider the coordinate transformation {a;} = [<^]{ii} or more explicitly
for the two degree of freedom system in Figure 1:
Thus
1
2
‘l -1
1 1
(5)
(6)
and enforcing the first mode shape (1, 1) in physical space should give a
second modal displacement of zero.
1
2
1 -1
1 1
1
1
(7)
So if an excitation is applied which causes the nonlinear system to vibrate
in its first linear mode shape, the response will be composed only of ui and
the influence of the coupled mode, U2, will then have been eliminated.
The method must therefore derive a force pattern which will cause the
system to vibrate in one of its linear mode shapes. It has been shown
1101
in a previous paper [3] that if the response contains harmonics then the
force pattern must also contain harmonics in order to control the harmonic
content of the response. In theory, the responses will be an infinite series
of harmonics, but this series is truncated in this case of a cubic stiffness
nonlinearity to include only the fundamental and third harmonic terms.
The physical input forces will thus be of the form:
fi(t) = Fii cos(a;ea;t 4- ^ii) 4- F13 cos(a;ea;t + (^13) (8)
f2{t) = F21 C0s(a;ea;t + (j>2l) 4* F23 COS{uJext + fe) (9)
where is the excitation frequency. Parameters for these force patterns
may then be chosen such that only mode one is excited.
OPTIMISATION APPROACH
In general, no a priori model of the system exists so an optimisation routine
is used to determine the force pattern parameters required to maximise the
contribution of the mode of interest. The objective function, the quantity
that the optimisation routine seeks to minimise, must be representative
of the deviation of the response from the target linear mode shape. The
objective function, F, that was chosen in this case was based on the vector
norm [4] of the two physical responses, Xi and X2, and is shown below:
where and 02 are elements of the mode shape vector for the target
mode. This summation is carried out over one cycle of the fundamental
response. The number of data points per cycle is npts and Xki the kth
response at the itk sample. This objective function allows the response
to contain harmonics and can be extended to more degrees of freedom by
choosing a reference displacement and subtracting further displacements
from it. The Variable Metric optimisation method [5] was used in this
work as it has been found to produce the best results for simulated data.
The application of this method to a two degree of freedom system such
as that shown in figure 1 is detailed in [6]. Optimised force patterns are
obtained at several levels of input amplitude. These force patterns are then
applied and the Restoring Force method is used to curve fit the resulting
modal displacement and velocity time histories to give the direct linear and
nonlinear coefficients for the mode of interest.
1102
EXPERIMENTAL IMPLEMENTATION
The simulated application of this method assumed that certain parame¬
ters were known. In order to carry out an identification of an experimental
structure, these parameters must be measured or calculated. Some pro¬
cessing of experimental data is necessary in order to apply the Restoring
Force method. The restoring force of a system can be expressed for a single
degree of freedom system as:
h{x, x) = f(t) - mx (11)
where h{x, x) is the restoring force and f{t) the input force. A similar ex¬
pression applies to the modal restoring force for an isolated mode. Thus the
input force, acceleration, velocity and displacement must be calculated at
the each time instant. A similar expression applies to the modal restoring
force for an isolated mode. In the experimentaJ case it is usual to measure
acceleration and input force; the remaining two states must therefore be
obtained by integration of the acceleration time history. Frequency domain
integration [7] was used for this purpose. High pass filtering was used to
remove any low frequency noise which can be amplified by this type of
integration. Several methods have been suggested for estimating modal
mass, but in this study a method developed by Worden and Tomlinson [8]
was used. An estimate for the modal mass is obtained and then an error
term is included in the curve fit which will iteratively yield a more accu¬
rate estimate of the mass. Generally the mass value will converge after one
iteration.
The objective function used in the simulations was calculated from the
displacement time histories. In the experimental case, acceleration was
used rather than displacement as it was considered that using ’raw’ data
would be quicker and give less opportunity for error. In the simulated
case, the system parameters were known a priori so the modal matrix
of the underlying linear system could be calculated. For most types of
nonlinearity the response of the system at low input force levels will be
dominated by linear terms. Normal mode tuning [1] was therefore applied
at low force levels to yield an approximation to the modal matrix of the
underlying linear system.
A quality indicator to give some idea of the effectiveness of the optimisa¬
tion performed would be advantageous. Results corrupted by background
noise, for example, could then be discarded. A perfect optimisation will
occur when the ratio of measured accelerations exactly matches the mode
shape ratio specified for the mode of interest. Thus a least squares fit of
the sampled accelerations was carried out over a cycle of the fundamental
frequency and the percentage error of the measured mode shape to the
1103
required mode shape was calculated. This percentage error will indicate
whether the optimisation has been successful.
To assess the accuracy of the parameters estimated using this method,
an identification was carried out using a conventional Restoring Force ap¬
proach in physical space. A band limited random excitation was used, and
the physical data processed and curve fitted. The physical parameters were
then transformed to modal space. The direct linear and nonlinear param¬
eters for modes one and two are shown in table 1. It should be noted that
although this conventional Restoring Force approach is possible for this
two degree of freedom system, it will not generally be possible since the
number of terms in the curve fit increases dramatically when different t3q)es
of nonlinearity and more degrees of freedom are included. It is carried out
in this case as a means of validating the proposed method.
EXPERIMENTAL SETUP
The rig constructed consisted of two masses on thin legs connected in series
by a linear spring, each mass being driven by a shaker. A cubic nonlin¬
earity was introduced between the first mass and ground using a clamped-
clamped beam attached at the centre which will yield a cubic stiffness for
large deflections [9], A schematic diagram of the rig is shown in figure 2.
The force input by each shaker was measured using a force gauge and the
acceleration of each mass was measured using an accelerometer in the po¬
sitions also shown in figure 2, Acceleration and force data were acquired
using a multiple channel acquisition system, the optimisation routine was
carried out on line.
RESULTS
Normal mode tuning of the rig gave natural frequencies of 20.67 Hz and
24.27 Hz and a modal matrix of:
3.87 5.03
5.52 -3.27
(12)
The excitation frequency was chosen to be slightly lower than the natural
frequency of the mode of interest in order to avoid the problems associated
with force drop out which are worst at the natural frequency. For each
mode optimisation was performed at three input force levels, the highest
level was as high as possible so as to excite the nonlinearity strongly. The
optimisation routine was carried out using the voltage input into the signal
generator as the variable. The force input into the structure was measured
1104
for use in the Restoring Force identification but was not used in the opti¬
misation as it is not directly controllable. The details of the optimisation
for each force level are presented in table 2. The optimised forces and re¬
sulting accelerations for mode one are shown in figure 3 and figure 4. The
acceleration data for the optimised force patterns were then integrated and
the modal restoring force for the mode of interest calculated. The initial
estimate of modal mass for the calculation of the modal restoring force was
tahen from a previous paper [10] in which the rig was identified using a
using a physical parameter identification method. The mass was estimated
in this paper to be 2.62 kg, this physical mass will then be equal to the
modal mass since the modal matrix was normalised to be orthonormal.
The restoring force data was then transformed to modal space. The modal
restoring force surface obtained using optimised force inputs for mode one
is shown in figure 5 and a stiffness section through this surface is shown in
figure 6. The restoring force time histories were then curve fitted against
modal velocity and displacement.
The estimated parameters for mode one axe shown in table 3. It can be
seen that they do not compare very well with those estimates obtained
using the band limited random excitation. It was suspected that at lower
excitation levels the estimates were being distorted by linear dependence
[11]. Linear dependence is a problem which occurs when curve fitting a
harmonic response from a linear system; the equations of motion may be
identically satisfied by mass and stiffness terms modified by an arbitrary
constant. This condition is avoided by the harmonic terms introduced into
the response by nonlinearity. The curve fit was thus repeated using only the
data obtained from the highest level of excitation; the estimates obtained
are shown in table 4. It can be seen that the linear parameter estimates
now agree well with the band limited random results. The estimates for
the cubic stiffness coefficient do not appear to agree so well. The standard
deviation on the cubic stiffness derived from the band limited random
excitation is approximately a third of the value of the parameter itself. The
uncertainty on this parameter occurs because the nonlinearity is not very
strongly excited by this type of excitation. A stiffness section through the
restoring force surface, figure 7, shows little evidence of a cubic stiffness
component. If a higher level of excitation were possible then a better
estimate may be achieved.
The identification w^ repeated for mode two. The restoring force surface
obtained and a stiffness section through it are shown in figure 8 and figure 9.
It can be seen from the stiffness section that the nonlinearity is not very
strongly excited. The estimated direct modal parameters are shown in
table 5. It can be seen that these results agree quite well with those
obtained using band limited random excitation. It is considered that the
discrepancy between the two sets of results, in particular the mass and
1105
stiifiiess estimates, is again due to linear dependence.
CONCLUSIONS
An extension of the force appropriation method has been proposed for
nonlinear systems. In this method, an optimisation routine is used to
determine the force patterns which will excite a single mode nonlinear
response . The direct linear and nonlinear modal parameters can then be
estimated from a curve fit of the modal restoring force surface. The method
was applied to an experimental two degree of freedom system whose modes
were coupled in linear modal space by a spring grounded nonlinearity.
A conventional restoring force identification was performed using a band
limited random signal for comparison. The parameters estimated from the
single mode responses were found to agree quite well with those from the
band limited random tests.
REFERENCES
1. Holmes P., Advanced Applications of Normal Mode Testing, PhD
Thesis, University of Manchester 1996.
2. Hadid M.A. and Wright J.R., Application of Force State Mapping
to the Identification of Nonlinear Systems Mechanical Systems and
Signal Processing, 1990, 4(6), 463-482
3. Atkins P.A., Wright J.R., Worden K., Manson G.M. and Tomlinson
G.R., Dimensional Reduction for Multi Degree of Freedom Nonlinear
Systems, International Conference on Identification in Engineering
Systems 1996, 712-721
4. Kreyszig E., Advanced Engineering Mathematics, Wiley, 6th Edition
5. Press W.H., Teukolsky S.A., Vettering W.T. and Flannery B.P., Nu¬
merical Recipes in Fortran Cambridge University Press, 6th Edition
6. Atkins P.A. and Wright J.R., An Extension of Force Appropriation
for Nonlinear Systems Noise and Vibration Engineering, Proceedings
of ISMA21(2), 915-926, 1996
7. Worden K., Data Processing and Experiment Design for the Restor¬
ing Force Method, Part I: Integration and Differentiation of Measured
Time Data, Mechanical Systems and Signal Processing, 4(4) 295-319,
1990
1106
8. Ajjan Al-Hadid M., Identification of Nonlinear Dynamic Systems us¬
ing the Force State Mapping Technique, PhD Thesis, Queen Mary
College, University of London, 1989
9. Storer D.M., Dynamic Analysis of Nonlinear Structures Using Higher
Order Frequency Response Function, PhD Thesis, University of Manch¬
ester, 1991
10. Atkins P. and Worden K., Identification of a Multi Degree of Freedom
Nonlinear System, Proc. of IMAC XV, 1997
ACKNOWLEDGEMENTS
This work was supported by E.P.S.R.C. under research grant number
GR/J48238 at the University of Manchester
Modal parameter
Mode one
Mode two
k (N/m)
c (Nm/s)
/3(N/m^)
m (kg)
4.87 X 10^
10.11
3.83 X 10®
2.60
6.49 X 10^
9.49
8.90 X 10*
3.06
Table 1: Direct modal parameters estimated from curve fit of band limited
random data
Low forcing
Medium forcing
High forcing
Fii (Volts)
1.0
2.0
3.0
F21 (Volts)
1.0
2.0
3.0
u)^ (Hz)
20.0
20.0
20.0
initial mode shape ratio
0.72
0.68
0.65
final mode shape ratio
0.70
0.70
0.70
target mode shape ratio
0.70
0.70
0.70
percentage error
0.14
0.03
0.71
Table ,2: Details of optimisation for mode one
Table 3: Direct modal parameters estimated from optimised responses
Model parameter Estimated parameter
k (N/m) 4.57 X 10^
c (Nm/s) 8.62
/?(N/m3) 6.81 X 10®
m (kg) _ _ 2^75 _
Table 4: Direct modal parameters estimated using high force level only
Model parameter
Estimated parameter
k (N/m)
c (Nm/s)
m (kg)
5.37 X 10^
10.16
2.45 X 10®
2.29
Table 5: Direct modal parameters estimated for mode two
Figure 1: Two degree of freedom system
1108
0 s 0.40008
Figure 3: Optimised forces for mode one at a high force level
Figure 4: Accelerations responses to optimised forces
Figure 5: Modal restoring force surface for mode one
Force (N)
Figure 7: Stiffness section through modal restoring force derived from ran¬
dom excitation for mode one
1111
Figure 8: Modal restoring force surface for mode two
Figure 9: Stiffness section through modal restoring force for mode two
1112
POWER FLOW TECHNIQUES II
THE OPTIMAL DESIGN OF NEAR-PERIODIC STRUCTURES TO
MINIMISE NOISE AND VIBRATION TRANSMISSION
R.S. Langley, N.S. Bardell, and P.M. Loasby
Department of Aeronautics and Astronautics
University of Southampton
Southampton SO 17 IBJ, UK
1. INTRODUCTION
An engineering structure is said to be of "periodic" construction if a basic
structural unit is repeated in a regular pattern . A beam which rests on
regularly spaced supports is one example of a one-dimensional periodic
structure, while an orthogonally stiffened cylinder is an example of a two-
dimensional periodic structure. It has long been known that perfectly periodic
structures have very distinctive vibration properties, in the sense that "pass
bands" and "stop bands" arise: these are frequency bands over which elastic
wave motion respectively can and cannot propagate through the structure [1,2].
If the excitation frequency lies within a stop band then the structural response
tends to be localised to the immediate vicinity of the excitation source.
Conversely, if the excitation frequency lies within a pass band then strong
vibration transmission can occur, and it is generally the case that the resonant
frequencies of the structure lie within the pass bands.
Much recent work has been performed concerning the effect of random
disorder on a nominally periodic structure (see for example [3-5]). It has been
found that disorder can lead to localisation of the response even for excitation
which lies within a pass band, and this reduces the propensity of the structure
to transmit vibration. This raises the possibility of designing disorder into a
structure in order to reduce vibration transmission, and this possibility was
briefly investigated in reference [6] for a one-dimensional periodic waveguide
which was embedded in an otherwise infinite homogeneous system. The
present work extends the work reported in reference [6] to the case of a finite
near-periodic beam system, which more closely resembles the type of
optimisation problem which is. likely to occur in engineering practice. The
beam is taken to have N bays, and the design parameters are taken to be the
individual bay lengths. Both single frequency and band-limited excitation are
considered, and two objective functions are investigated: (i) the response in a
bay which is distant from the applied loading (minimisation of vibration
transmission), and (ii) the maximum response in the structure (minimisation
of maximum stress levels). In each case the optimal configuration is found by
employing a quasi-Newton algorithm, and the physical features of the resulting
design are discussed in order to suggest general design guidelines.
1113
2. ANALYTICAL MODEL OF THE NEAR-PERIODIC BEAM
2.1 Calculation of the Forced Response
A schematic of an N bay near-periodic beam structure is shown in Figure 1.
The structure is subjected to dynamic loading, and the aim of the present work
is to find the optimal design which will minimise a prescribed measure of the
vibration response. No matter what type of optimisation algorithm is
employed, this type of study requires repeated computation of the system
dynamic response as the design parameters are varied, and it is therefore
important to employ an efficient analysis procedure. In the present work the
h-p version of the finite element method (FEM) is employed: with this
approach the structure is modelled as an assembly of elements which have
both nodal and internal degrees of freedom. Each element has two nodes and
the nodal degrees of freedom consist of the beam displacement and slope; the
internal degrees of freedom are generalised coordinates which are associated
with a hierarchy of shape functions which contribute only to the internal
displacement field of the element. The internal shape functions used here are
the K-orthogonal Legendre polynomials of order four onwards - full details of
the present modelling approach are given in reference [7].
For harmonic excitation of frequency co the equations of motion of the
complete beam structure can be written in the form
where M and K are the global mass and stiffness matrices (assembled from the
individual element matrices taking into account the presence of any mass or
spring elements and allowing for constraints), q contains the system
generalized coordinates, F is the generalized force vector, and r| is the loss
factor, which in the present study is taken to be uniform throughout the
structure.
Equation (1) can readily be solved to yield the system response q. In the
present work it is convenient to use the time averaged kinetic and strain
energies of each of the N bays as a measure of the response - for the nth bay
these quantities can be written as and say, where
r,=(£0V4)9;>„?„. (2.3)
Here and K„ are the mass and stiffness matrices of the nth bay, and q„ is
the vector of generalized coordinates for this bay.
Many of the physical features of the forced response of a near-periodic
structure can be explained in terms of the free vibration behaviour of the
1114
associated perfectly periodic structure. The following section outlines how the
present finite element modelling approach can be used to study the pass bands
and stop bands exhibited by a perfect periodic structure.
2.2 Periodic Structure Analysis
The finite element method described in section 2.1 can be applied to a single
bay of a perfectly periodic structure to yield an equation of motion in the form
Dq=F, D=-coW--(l+/r|)ii:, (^’5)
where the matrix D is referred to as the dynamic stiffness matrix. In order to
study wave motion through the periodic system it is convenient to partition D,
q and F as follows
r
Du
ft:
D=
D„
D,r
. F=
0
Dr,
F„
V V
(6-8)
where L relates to the coordinates at the left most node, R relates to those at
the right most node, and / relates to the remaining "internal" coordinates.
Equations (4-8) can be used to derive the following transfer matrix relation
between the displacements and forces at the left and right hand nodes
V /
V /
(9,10)
Equation (9) can now be used to analyze wave motion through the periodic
system: such motion is governed by Bloch’s Theorem, which states that
i^L F i)=exp{-ie-b){qii -F^) where 8 and 5 are known respectively as the phase
and attenuation constants. A pass band is defined as a frequency band over
which 6=0, so that wave motion can propagate down the structure without
attenuation. It follows from equation (9) that
(T-Ie
-ie-6'
( \
0
F,
io
V
V y
(11)
SO that 8 and 6 can be computed from the eigenvalues of T, thus enabling the
pass bands and stop bands to be identified.
1115
2.3 Optimisation Procedure
Equations (l)-(3) enable the forced response of the system to be calculated for
any prescribed set of system properties. The aim of the present analysis is to
compute the optimal set of system properties for a prescribed design objective,
and in order to achieve this equations (l)-(3) are evaluated repeatedly as part
of an optimisation algorithm. As an example, it might be required to minimise
the kinetic energy of bay N by changing the various bay lengths. In this case
equations (l)-(3) provide the route via which the objective function (the kinetic
energy in bay N) is related to the design parameters (the bay lengths), and the
optimisation algorithm must adjust the design parameters so as to minimise the
objective function. The optimisation process has been performed here by using
the NAg library routine E04JAF [8], which employs a quasi-Newton algorithm.
This type of algorithm locates a minimum in the objective function, although
there is no indication whether this minimum is the global minimum or a less
optimal local minimum. The probability of locating the global minimum can
be increased significantly by repeated application of the NAg routine using
random starts, i.e. random initial values of the design parameters. Numerical
investigations have led to the use of 30 random starts in the present work.
3. NUMERICAL RESULTS
3.1 The System Considered
The foregoing analysis has been applied to a beam of flexural rigidity El, mass
per unit length m, and loss factor r|=0.015, which rests on A+1 simple
supports, thus giving an A’-bay near-periodic system. The design parameters
are taken to be the bay lengths (i.e. the separation of the simple supports), and
the design is constrained so that the length of any bay lies within the range
0.9L^,<1.1L, where L, is a reference length. A non-dimensional frequency
Q is introduced such that 0.=(oL,N{m/Er), and the non-dimensional kinetic and
strain energies of a bay are defined as T,'-T,^(EI/L^^\F\^) and
U,'~U„iEI/L,^\F\'^) where F is the applied point load. As discussed in the
following subsections, two objective functions are considered corresponding
to minimum vibration transmission and minimum overall response. In all
cases the excitation consists of a point load applied to the first bay and the
response is averaged over 1 1 equally spaced point load locations within the
bay. For reference, the propagation constants for a periodic system in which
all the bay lengths are equal to L, are shown in Figure 2 - the present study is
focused on excitation frequencies which lie in the range 23<f2<61, which
covers the second stop band and the second pass band of the periodic system.
3.2 Design for Minimum Vibration Transmission
In this case the objective function is taken to be the kinetic energy in bay N,
so that the aim is to minimise the vibration transmitted along the structure.
Three types of loading are considered; (i) single frequency loading with Q=50,
1116
which lies within the second pass band of the ordered structure; (ii) band-
limited loading with 40<Q<60, which covers the whole of the second pass
band; (iii) band-limited loading with 23<Q<61, which covers the whole of the
second stop band and the second pass band.
Results for the optimal design under single frequency loading are shown in
Table 1; in all cases it was found that the bay lengths were placed against
either the upper bound (U=1.1L,) or the lower bound (L=0.9L,), and significant
reductions in the energy level of bay N were achieved. In this regard it should
be noted that the dB reduction quoted on Table 1 is defined as -101og(r;v/^;^r)
where is the kinetic energy in the final bay of the ordered system. The
optimal designs shown in Table 1 all tend to consist of a bi-periodic structure
in which the basic unit consists of two bays in the configuration LU. The pass
bands and stop bands for this configuration are shown in Figure 3, and further,
for the optimal 12 bay system is shown in Figure 4 over the frequency
range 0<Q<250. By comparing Figures 3 and 4 it is clear why the selected
design is optimal - the new bi-periodic system has a stop band centred on the
specified excitation frequency Q=50. It can be seen from Figure 4 that the
improvement in the response at the specified frequency 0=50 is accompanied
by a worsening of the response at some other frequencies.
Results for the optimal design under band-limited excitation over the range
40<Q<60 are shown in Table 2. In some cases two results are shown for the
optimised "Final Energy": in such cases the first result has been obtained by
forcing each bay length onto either the upper (U) or lower (L) bound, while
the second result has been obtained by using the NAg optimisation routine.
If only one result is shown then the two methods yield the same optimal
design. The "bound" result is easily obtained by computing the response under
each possible combination of U and L bay lengths - this requires 2^ response
calculations, which normally takes much less CPU time than the NAg
optimisation routine. It is clear from Table 2 that the additional improvement
in the response yielded by the full optimisation routine is minimal for this
case. The response curve for the 12-bay system is shown in Figure 5, where
it is clear that a significantly reduced response is achieved over the specified
frequency range; as would be expected an increase in the response can occur
at other frequencies. It is interesting to note that most of the optimal designs
shown in Table 2 lack symmetry - however, it follows from the principle of
reciprocity that a design which minimises vibration transmission from left to
right will also minimise transmission from right to left. It should therefore be
possible to "reverse" the designs without changing the transmitted vibration
levels. This hypothesis is tested in Figure 6 for a 12 bay structure - the figure
shows the energy distribution for the optimal design UUULUULLLLLU and
for the reversed design ULLLLLUULUUU. Although the detailed distribution
of energy varies between the two designs, the energy levels achieved in bay
1117
12 are identical, as expected.
Results for the optimal design under wide-band excitation 23<Q<61 are shown
in Table 3, and the response curve for the 12-bay optimised system is shown
in Figure 7. The form of optimal design achieved is similar to that obtained
for the narrower excitation band 40<Q<60, although there are detailed
differences between the two sets of results. In each case there is a tendency
for a group of lower bound bays (L) to occur in the mid region of the
structure, and a group of upper bound bays (U) to occur at either end. This
creates an "impedance mismatch" between the two sets of bays, which
promotes wave reflection and thus reduces vibration transmission along the
structure. By comparing Tables 1-3, it is clear that the achievable reduction
in vibration transmission reduces as the bandwidth of the excitation is
increased.
3.3 Design for Minimum ''Maximum” Strain Energy
In this case the strain energy U„ of each bay is computed and the objective
function is taken to be the maximum value of U„. As a design objective, this
procedure can be likened to minimising the maximum stress in the structure.
As in the previous section the three frequency ranges Q=50, 40<n<60, and
23<Q<61 are considered, and the present study is limited to systems having
9,10, 11, or 12 bays; the optimal designs achieved are shown in Table 4.
Considering the single frequency results (Q.=50) shown in Table 4, it is clear
that a large dB reduction is achieved only for those systems which have an
even number of bays; furthermore, the optimal energy obtained has the same
value (0.0297) in all cases. This can be explained by noting that for an odd
number of bays the frequency Q=50 lies near to an anti-resonance of the
ordered structure, whereas a resonance is excited for an even number of bays -
this feature is illustrated in Figure 8 for the 12 bay structure. The repeated
occurrence of the optimal energy 0.0297 arises from the fact that the initial bay
pattern ULLLUUU occurs in all four designs - it has been found that this
pattern causes a vibration reduction of over 20dB from bay 1 to bay 8, so that
the response in bay 1 (the maximum response) is insensitive to the nature of
structure from bay 8 onwards.
The optimal "bounded" designs arising for band-limited excitation either tend
to be of the "UL" bi-periodic type or else nearly all the bays are assigned the
same length. However it should be noted that in all cases the design produced
by the NAg optimisation routine offers an improvement over the "bounded"
design, particularly for the wide-band case (23<n<61). It is clear from Table
4 that the achieved reduction in strain energy reduces as the bandwidth of the
excitation is increased.
1118
4. CONCLUSIONS
The present work has considered the optimal design of a near-periodic beam
system to minimise vibration transmission and also maximum stress levels.
With regard to vibration transmission it has been found that very significant
reductions in transmission are achievable with relatively minor design changes.
The optimum design normally involves placing the design parameters (the bay
lengths) on the permissible bounds, and this means that a simple design search
routine can be used in preference to a full optimisation algorithm. With regard
to minimum stress levels, it has been found that the optimal design for wide¬
band excitation is not normally a "bounded" design, and thus use of a full
optimisation algorithm is preferable for this case. For both vibration
transmission and maximum stress levels, the benefits obtained from an optimal
design decrease with increasing excitation bandwidth, but nonetheless very
significant reductions can be obtained for wide-band excitation.
REFERENCES
1. S.S. MESTER and H. BENAROYA 1995 Shock and Vibration 2, 69-
95. Periodic and near-periodic structures.
2. D.J. MEAD 1996 Journal of Sound and Vibration 190, 495-524. Wave
propagation in continuous periodic structures: research contributions
from Southampton 1964-1995.
3. C.H. HODGES 1982 Journal of Sound and Vibration 82, 411-424.
Confinement of vibration by structural irregularity.
4. D. BOUZIT and C. PIERRE 1992 Journal of Vibration and Acoustics
114, 521-530. Vibration confinement phenomena in disordered, mono-
coupled, multi-span beams.
5. R.S. LANGLEY 1996 Journal of Sound and Vibration 189, 421-441.
The statistics of wave transmission through disordered periodic
waveguides.
6. R.S. LANGLEY 1995 Journal of Sound and Vibration 188, 717-743.
Wave transmission through one-dimensional near periodic structures:
optimum and random disorder.
7. N.S. BARDELL, R.S. LANGLEY, J.M. DUNSDON and T. KLEIN
1996 Journal of Sound and Vibration 197, 427-446. The effect of
period asymmetry on wave propagation in periodic beams.
8. ANON 1986 The NAg Fortran Workshop Library Handbook - Release
1. Oxford: NAg Ltd.
1119
TABLE 1
Optimal design of 1-D beam structure, to minimise energy transmission, D.=50.
Original Energy; Non-dimensional kinetic energy in bay N of the periodic structure.
Final Energy; Non-dimensional kinetic energy in bay N of the optimised structure.
No. of
Optimal Pattern
Original
Final
Reduction
Bays, N
Energy
Energy
(dB)
4
UULU
0.276E 1
0.804E-3
35.348
ULULU
0.609E-1
0.179E-3
■S
UULULULU
0.674E 0
0.613E-5
ULULULULU
0.564E-1
0.135E-5
46.216
1—
10
UULULULULU
0.424E 0
0.532E-6 ^
11
ulululululu
0.535E-1
0.n7E-6
56.604
12
UULULULULULU
0.289E 0
0.461E-7
67.966
13
ULULULULULULU
0.502E-1
O.lOlE-7
66.950
16
UULULULULULULULU
0.154E0
0.346E-9
_ _ —
86.484
17
ululululululululu
0.43 lE-1
0.761E-10
87.529 1
1120
TABLE 2
Optimal design of 1-D beam structure, to minimise energy transmission, 40<Q<60.
Original Energy: Non-dimensional kinetic energy in bay N of the periodic structure.
Final Energy: Non-dimensional kinetic energy in bay N of the optimised structure.
No. of
Optimal pattern
Original |
Final
Reduction
Bays, N
Energy
Energy
(dB)
■■
ULLU
0.670E 0
0.103E-1
18.112
5
ULLLU
0.63 IE 0
0.735E-2
19.338
0.711E-2
19.482
6
UULLLU
0.221E-2
22.407
7
ULLUULU
0.463E 0
0.171E-2
24.335
8
UULLLLLU
0.430E 0
0.966E-3
26.487
0.914E-3
26.725
9
UUULLLLLU
0.444E 0
0.341E-3
31.142
10
UUUULLLLLU
0.449E 0
0.192E-3
33.681
0.189E-3
33.758
11
ULLUUUULLLU
0.291E0
0.821E-4
35.504
12
UUULUULLLLLU
0.201E0
0.352E-4
37.558
13
1 ULUUUUULLLLLU
0.199E0
0.153E-4
41.148
1121
TABLE 3
Optimal design of 1-D beam structure, to minimise energy transmission, 23<Q<6L
Original Energy; Non-dimensional kinetic energy in bay N of the periodic structure.
Final Energy: Non-dimensional kinetic energy in bay N of the optimised structure.
6
LLLLUU
7
LLLULUU
8
LLLLLUUU
9
ULLLLLLUU
0.494E 0
0.183E0
0.175E0
0.648E-2
0.246E-2
0.180E-2
III »
UUULLLLLUU
0.105E0
0.277E-3
2.762
3.916
18.821
18.715
19.878
21.868
25.787
11
UUULLLLLLUU
0.105E0
0.776E-4
31.313
12
UlTULLLLLLLUU
0.166E0
0.526E-4
34.991
13
UUULLLLLLULUU
0.973E-1
0.282E-4
35.379
14
UUUULLLLLLULUU
0.581E-1
0.122E-4
36.778
TABLE 4
Optimal design ofl-D beam structure, to minimise "maximum” strain energy.
Bay No.: Bay in which the optimal minimum “maximum” non-dimensional strain energy occurs
Original Energy: Initial “maximum” non-dimensional bay strain energy of the periodic structure.
Final Energy: Non-dimensional strain energy in bay N’ of the optimised structure.
No of
Optimal
Original
Final
Bay
Reduction
Bays, N
Pattern
Energy
Energy
No., N’
(dB)
Q.=50
9
ULLLUUUUL
0.667E-1
0.297E-1
1
3.514
0.296E-1
1
3.528
10
ULLLUUUULU
0.540E 0
12
ULLLULUULULU
0.404E 0
0.297E-1
1
11.336
0.296E-1
1
11.351
40<Q<60
9
UUUUULULU
0.486E 0
0.710E-1
1
8.354
0.449E-i
1-2
10.344
10
ULULULULLL
0.606E 0
0.643E-1
1
9.743
0.45 IE- 1
1-2
11.283
11
ULULULULLUU
0.456E 0
0.682E-1
I
8.252
0.425E-1
1-2
10.306
12
UUUIJUUUXJUUUL
0.332E 0
0.550E-1
2
7.808
0.412E-1
1-2
9.062
23<n<61
9
LLLLLLLLL
0.234E 0
0.203E 0
1
0.617
0.979E-1
1
3.784
n
LLLLLLLLLL
0.200E 0
0.178E0
1
0.506
■I
0.95 IE- 1
1-2
3.228
11
UUUUUUUUULL
0.198E0
0.193E0
1
O.lll
0.910E-1
1-2
3.376
12
UUUUUOUIJULUU
0.314E0
0.182E0
I
2.369
0.803E-1
1-2
5.922
1123
Phase £ AUenualion
Figure 1; A simply supported periodic
Itlijsj
kinelic.enorgy in bay 12, T,
1128
EFFECTS OF GEOMETRIC ASYMMETRY ON VIBRATIONAL
POWER TRANSMISSION IN FRAMEWORKS
J L Homer
Department of Aeronautical and Automotive Engineering
and Transport Studies, Loughborough University
Loughborough, Leics , LEll 3TU, UK
ABSTRACT
Many sources, such as machines, are installed on supports, or frameworks,
constmcted from beam-like members. It is desirable to be able to predict
which wave types will be present at particular points in the support structure.
By using the concept of vibrational power it is possible to compare the
contributions from each wave type. Wave motion techniques are used to
determine the expressions for vibrational power for each of the various wave
types present. The results from the analysis show the amount of vibrational
power carried by each wave type and the direction of propagation.
Consideration is given to the effect on the vibrational power transmission of
introducing misalignment of junctions in previously symmetric framework
structures. By splitting a four beam junction in to, say, a pair of three beam
junctions separated by a small distance, it is possible to establish the effects of
separating the junctions on the various transmission paths. Unlike other
techniques using vibrational power to analysis frameworks, the model keeps
the contributions from each of the various wave types separate. This allows
decisions to be made on the correct vibrational control techniques to be
applied to the structure.
INTRODUCTION
When attempting to control vibration levels transmitted from a machine
through the various connections to the structure upon which it is mounted, it is
desirable to be able to identify and quantify the vibration paths in the stmcture.
Often large machinery installations are installed on frameworks consisting of
beam like members. These frameworks are then isolated from the main
structure. Simple framework models are also used in the initial design stages
of automotive body shell structures to determine dynamic responses.
If the dominant transmission path in the framework is identified it is
possible to reduce vibration levels by absorbing the mechanical energy along
the propagation path in some convenient manner. By utilising the concept of
vibrational power it is possible to quantitatively compare the wave type
contributions to each transmission patL In order to predict vibrational power
transmission in a framework, it is necessary to identify the wave amplitude
reflection and transmission coefficients for each joint in the structure. Lee and
Kolsky [1] investigated the effects of longitudinal wave impingement on a
junction of arbitrary angle between two rods. Similarly Doyle and Kamle [2]
examined the wave amplitudes resulting from a flexural wave impinging on
the junction between two beams. By using the reflection and transmission
coefficients for different joints, it is possible to predict the vibrational power
associated with flexural and longitudinal waves in each section of the
1129
framework. Previous investigations [3,4] have considered the effects of bends
and junctions in infinite beams. This work was extended to consider the finite
members which constitute frameworks [5]. Unlike other techniques [6, 7]
utilising energy techniques to analyse frame-works, the technique produces
power distributions for each wave type present in the structure. By comparing
the results for each wave type, it is possible to apply the correct methods of
vibration control.
The technique is used to investigate the effect of geometric asymmetry
on the vibration transmission, due to steady state sinosodial excitation, in a
framework structure similar to, say, those used in the automotive industry
(figure 1). By splitting a four beam junction into a pair of three beam
junctions separated by a known distance, it is possible to establish the effect of
junction separation on the dominant transmission paths. The investigation
presented is limited to one dimensional bending waves and compressive waves
only propagating in the structure. To consider the addition of other wave
types ie. torsional waves and bending waves in the other plane, the analysis
presented here for the junctions should be extended as indicated by Gibbs and
Tattersall [3].
TRANSMITTED POWER IN A UNIFORM BEAM
For flexural wave motion, consider a section of a uniform beam carrying a
propagating flexural wave. Two loads act on this beam element, the shear
force and the bending moment. It is assumed that the flexural wave can be
described by using Euler-Bemoulli beam theory, so that the displacement can
be expressed as
W(x,t) = Af sin (cot-kfx),
the shear force acting on a section as
S = Eia3W/ax3,
and the bending moment on the section as
B = El 02 W/ax2.
Then the instantaneous rate of working X at the cross-section is given
by the sum of two terms (negative sign merely due to sign convention).
0t 0x0t 0X^ 0X^ 3x0t
The time averaged power
{P)f = (1/T) I X dt then is given by (P)f = Elkf coAf
I (1)
For longitudinal wave motion consider a section of a uniform beam
with a longitudinal wave propagating through the beam
U (x, t) = Ai sin (0)t-kix)
1130
The instantaneous rate of working X is then
X= -EA(au/ax)u
and the time averaged power is
Xdt = rEA0)kiA|
(2)
If dissipation is present in the structure, the modulus of elasticity may
be considered to be a complex quantity
= E(l + iTi)
where represents the loss factor of the material, present due to
inherent material damping.
The displacement of a beam at a distance x from the source, due to
flexural wave motion may now be considered to be, assuming that
material damping is small.
1 ^
-kri-
W = Afe 4
and the resulting time averaged power is given by
(P}f = EIcok^e-'‘'’i2 Af
(3)
The above reduces to equation (1) at the source.
Similarly, the displacement of beam, due to longitudinal wave motion
may be considered to be
T ^
-kiTi-
U = Aie ^ sin(cot-kix)
and the resulting time averaged longitudinal power may be rewritten as
(P),=iEAcok,e-‘''^’‘A?
(4)
WAVE TRANSMISSION THROUGH A MULTI BRANCH JUNCTION
Consider a four branch junction as shown in figure 2. Assuming only flexural
and longitudinal waves propagating in the structure, the displacements of Arm
1 will be, where A4 represents the impinging flexural wave arriving from
infinity.
Wi(x,t) = (Aie^f’"" + A3e‘^fi'' + A4e e‘“^
(5)
1131
(6)
U,(x,t) = (Aae''‘‘i’‘)e‘“‘
Similarly for arms 2 to 4 the displacement will be,
where ^ cos 0n and n is the beam number
W„(v„,t) = )e‘“‘
(V)
(8)
Here A3, A4, 64^ are travelling flexural wave amplitudes', Ai and
B2n are near field wave amplitudes and Aa and are travelling
longitudinal wave amplitudes.
In previous work [2] in this field a theoretical model was used in which
it was assumed that the junction between the beams was a rigid mass. The
mass or joint is modelled here as a section of a cylinder. This represents the
physical shape of most joints in practical systems. It has been shown [4] that
the joint mass has an insignificant effect on the reflected and transmitted
power for the range of values used in this work.
The joint mass Mj = pjTtL^J^/ 4, and the moment of inertia of the joint
isIj=ML2/8.
By considering the conditions for continuity and equilibrium at the
beam junction the following expressions may be written.
For each arm
For continuity of longitudinal displacement
L 3W
Ui=Un COsen-WnSinen + -^^sinen
For continuity of flexural displacement
W,=u„sinen+W„cos0„-| |^(l + cose„)
For continuity of slope
8W; ^ awn
1132
For the junction
Equilibrium of forces
' ' 2 ' ' 3x2 J
-vfp fax L32w„
1 1 I aVS 2 3v|;^
El Ai ^ + Mj ^ = i[E„ A„^cose„ +E„ I„^^sine„
3x J dt^ il^ 3V„ 3<
E T a^W, 3 r„, L3W|
El Ii — t^ + M; — T Wi- — — —
- ^ -I- iVi ; — y 1 “ “ ^T"
3x2 2 3x
n ;^TI ^ W
= Z En Ajj- ”Sin6n-EnIn 2*^ COS0n
1 I 5¥n
WAVE MOTION AT A FORCED OR FREE END
As indicated in figure 1 , the framework has one forced end and one free end.
Assuming the structure is only excited by a transverse harmonic force, the
boundary conditions are as follows:
at the forced end
E« = Pe^«t
3x2
EI^ = 0
3x2
EA^ = 0
Similarly at the free end the above boundary conditions apply with the
exception that
EI^ = 0
3x2
1133
POWER TRANSMISSION THROUGH A FRAMEWORK
The structure shown in figure 1 consits of one four-beam junction, two three-
beam junctions and four two-beam junctions. From the equations detailed in
the above two sections, it is possible to construct matrices of continuity and
equilibrium equations for sub structures. These may be combined to obtain
the overall matrix for the system. For the framework shown in figure 1, the
size of the overall matrix is 60x60. This matrix may be solved to obtain the
sixty unknown wave amplitude coefficients from which time averaged
transmitted power for each beam may be calculated using equations (3) and
(4).
Normalised nett vibrational power is then calculated at the centre of
each beam constituting the structure. Nett vibrational power may be
considered to be the difference between power flowing in the positive
direction and power flowing in the negative direction for each wave type.
Normalised nett power is considered to be nett power divided by total input
power. The input power to a structure may be calculated from the following
expression [8]
Input Power = ^ IFI IVI cos0
where 0 is the phase angle between the applied force and the velocity
of the structure at the forcing position.
Figures 3-6 show the nett normalised power in each arm of a
framework structure over the frequency range 0-lkHz excited by IN force,
whose material and geometric properties are given in Appendix 2. For the
results shown, angle 1 is 45® and angle x is 40® (or the ratio
= 0.89) and
L = y = 0.1m. Using these parameters the ratio of the length of beam No.6 to
beam No.4 is 0.12. The predicted flexural power is shown in figures 3 and 4
and from these it can be seen that the dominant transmission paths are arms 1
and 5, the forced and free arms. The transmitted power in arm 10 is next
dominant and comparable to arm 5 in the region 0-600Hz.
The response for ail other arms are small, typically less than 5% of
input power, with, as would be expected, arms 2 and 9 being approximately
identical in transmission properties.
Figures 5 and 6 show the nett normalised power for the longitudinal
waves in the structure. As the frequency range of interest corresponds to a
flexural Helmoltz number of 1 to 5 with L being the reference length, the
conversion of power from flexural to longitudinal waves is minimal. From the
figures it can be seen that beams 1, 5, 6 and 10 have identical transmission
characteristics, which would be expected at such large longitudinal
wavelengths. Significant longitudinal power is only observed in arms 3 and 8
in the frequency region 200-300Hz. This frequency region coincides with a
drop in the flexural power due to the structure being at resonance in that
region. It should be noted that power transmitted through arms 3 and 8 has
travelled through two junctions.
1134
EFFECT OF GEOMETRIC ASYMMETRY
By altering the ratio of angle 6i to angle 0x it is possible to alter the length of
beam 6 and hence move a pair of three arm junctions further or closer apart.
From the discussion in the previous section, it was seen, for the structure
under investigation, that the dominant flexural path, not surprisingly, is
through the centre of the structure, whilst the peaks in longitudinal power
occur in beams 3 and 8. Thus 0x was varied and the effect on transmission in
the dominant paths noted.
Figures 7-9 show flexural power for arms 5 and 10 and longitudinal
power for arm 8 for four values of 0x. The values chosen were 36°, 38.25°,
40° and 42.75° which are equivalent respectively to 0x over 0] ratios of 80%,
85%, 90% and 95%. Thus as 0x increases, the structure moves to being
symmetrical in nature. From figure 7, the increase in junction separation
decreases power in the frequency region 0-500Hz and increases it in the region
500-lkHz. In beam 10 (figure 10) the effect on the flexural power is reversed
with increase in junction separation leading to increased power below 500Hz
and decreased power above 500Hz. It should also be noted that increased
junction separation has little effect on the power below 250Hz. The effect was
also noted on all other beams which had both ends connected to a joint. It
may be concluded that at long flexural wavelengths the junction separation has
little effect with the impedance mis-match at the junctions being the important
criteria to effect transmission. It should also be noted that increasing power in
one arm ie. 5, causes a decrease in arms ie. 10, connected to it. An example of
the effect of junction separation on longitudinal power is shown in figure 9.
This shows nett normalised longitudinal power for arm 8 for the same
variation in 0x. Again minimal effect is seen at low frequencies, with
increased junction separation having different effects in different frequency
region. Increasing junction separation has little effect on the region between
200 and 300Hz when the longitudinal power was dominant. This would be
caused by the junction separation having little effect on the structures flexural
natural frequencies. Only by shifting those would the peaks in longitudinal
power by shifted in frequency.
CONCLUSIONS
Results are presented for normalised nett time average vibrational power for a
framework structure. The geometric symmetry of the structure is broken by
allowing one angle to decrease in value. The effects of varying the angle
change by up to 20% of its original value are investigated. Although the
results presented are for one example only, highlighted are the fact that
decreases in power in one part of the structure result in increases in power in
another part. Also shown was the effect of splitting a junction in to a pair of
junctions is minimal at low frequencies, or long wavelengths. From the results
of the analysis it is possible to establish frequencies and positions for
minimum power on the structure. Other configurations of framework
structure may be analysed by applying the equations presented.
1135
REFERENCES
1. J. P. LEE. and H. KOLSKY 1972 Journal of Applied Mechanics 39, 809-
813. The generation of stress pulses at the junction of two non-collinear
rods.
2. J. F. DOYLE and S. KAMLE 1987 Journal of Applied Mechanics 54,
136-140. An experimental study of the reflection and transmission of
flexural waves at an arbitrary T-Joint.
3. B. M. GIBBS and J. D. TATTERS ALL 1987 Journal of Vibration,
Acoustics, Stress and Reliability in Design, 109, 348-355. Vibrational
energy transmission and mode conversion at a corner junction of square
section rods.
4. J. L. HORNER and R. G. WHITE 1991 Journal of Sound and Vibration
147, 87-103. Prediction of vibrational power transmission through bends
and joints in beam-like structures.
5. J. L. HORNER 1994 Proceedings of the 5th International Conference on
Recent Advances in Structural Dynamics, SOUTHAMPTON UK, 450-
459. Analysis of vibrational power transmission in framework structures.
6. P. E. CHO and R. J. BERNHARD 1993 Proceedings of the 4th
International Congress on Intensity Techniques, SENLIS, France, 347-
354. A simple method for predicting energy flow distributions in frame
structures.
7. M. BESHARA and A. J. KEANE 1996 Proceedings of Inter-Noise '96,
LIVERPOOL, UK 2957-2962. Energy flows in beam networks with
complient joints.
8. R.J.PINNINGTON and R.G.WHITE 1981 Journal of Sound and Vibration
75, 179-197. Power flow through machine isolators to resonant and non-
resonant beams.
APPENDIX 1 - NOTATION
A
- Cross sectional area
Q
- Axial force
Af
- Amplitude of flexural wave
s
- Shear force
Ai
- Amplitude of longitudinal wave
T
- Time period
B
- Bending moment
t
- Time
E
- Young’s modulus
U
- Displacement due to
E*
- Complex Young’s modulus
longitudinal wave motion
F
- Excitation force
V
- Velocity
I
- Moment of inertia
w
- Displacement due to
flexural wave motion
Ij
Jw
- Moment of inertia of joint
X
- Instantaneous rate of
working
- Joint width
X
- Distance
- Loss factor
kf
- Flexural wave number
0n
- Angle of Arm n
ki
- Longitudinal wave number
Pj
- Joint density
L
- Joint length
M
- Moment force
Mj
n
P
- Joint mass
<}>
Phase angle
- Beam number
- Transverse force
¥n
- Distance along Arm n
1136
<P>f - Time averaged flexural power co - Frequency (rad/s)
<P>1 - Time averaged longitudinal power
APPENDIX 2 - MODEL PROPERTIES
33mm
6mm
5GN/m2
1180kg/m3
0.001
1
Beam Breadth
Beam Depth
Youngs Modulus
Density
Loss factor
Figure 1 : Framework Structure
iNett Normalised Power
Nett Normalised Power
Figure 5: Longitudinal Power - Beams 1-5
(Beam 1 - , Beam 2 . , Beam 3 . . Beam 4 - , Beam 5
Figure 6: Longitudinal Power - Beams 6-10
(Beam 6 - , Beam 7 . , Beam 8 . . Beam 9 - , Beam 10
1139
THE INFLUENCE OF THE DISSff ATION LAYER ON ENERGY
FLOW IN PLATE CONNECTIONS
Marek Iwaniec, Ryszard Panuszka
Technical University of Mining and Metallurgy,
Structural Acoustics and Intelligent Materials Group
30-059 Cracow, al. Mickiewicza 30, Poland
1. Introduction
Dynamic behaviour of mechanical strucmres may be modelled on the basis of
and with the help of mathematical apparams used in Statistical Energy Analysis (SEA)
[5] The method is especially useful to calculate the statistical approach vibroacoustical
energy flow in middle and high frequency range. With the help of a few parameters,
such'as- modal density, damping loss factor, coupling loss factor and the value of
input power, building linear equations set it is possible to describe the flow of
vibroacoustical energy in a complicated stmcture. There is also a possibility of quick
estimation of the influence of constmction method on the vibroacoustical parameters
of the whole set. In the following work an exemplary application of one of the most
frequently used software for calculating the flow of acoustic energy has been
presented- AutoSEA programme [1]. The aim of the work is practical modelling of
vibroacoustical energy flows through screw-connection of two plates and comparing
quantity results with experimental (outcome) measurements. Equivalent coupling loss
factor has been calculated for a group of mumally combined elements constimting a
construction fragment. A comparison between the measured results and the value of
coupling loss factor in linear joint (e.g. in welded one) has also been made. Using the
method of fmite elements, the influence of rubber separator thickness on the value of
the first several frequencies of free vibrations has been computed as well.
2. Physical model of plate connection
A connection of two perpendicular plates has been chosen for modelling the
flow of vibroacoustical energy in mechanical joints. Connection diagram is presented
in Figure 1. On the length of common edge the plates has been joined with anglesteel
by screws. A rubber separator (4) has been placed between the excited plate and the
anglesteel leg (3).
1143
Modelled stmcture
3. SEA model
In order to carry out the vibroacoustical analysis of the system using Statistical
Energy Analysis a model of the examined strucmre has been built. It has been
assumed that in every element of the construction only flexural waves propagate.
Every plate and the rubber layer have been modelled with just one appropriately
chosen subsystem. The anglesteel, however, has been modelled as a continuous
connection of two plates having the dimensions which correspond to the anglesteel
legs the plates themself being set at the right angle.
Fig. 2. SEA model of a system Fig- 3. Modal densities of the subsystems
1144
Using the SEA method we are able to describe the flow of the vibroacousdcal
energy in middle and high frequencies with an algebraic equation set. The exMined
system consists of five simple subsystems, of which only one is exited to vibration
with applied force. The flow of vibroacoustical energy m the model presented is
depicted with the following equation:
^ 1 tot
-^21
0
0
12
^ 2 tot
32
0
0
“'•123
^ 3 tot
-‘n43
0
0
“''134
h 4 tot
0
0
0
"'’145
0
tot
w,
0
^2 tot
0)
0
0
•
^3 tot
=
0
-^54
^4 tot
0
^ 5 tot
^5 tot
0
^ ^ 4- is a total coefficient of energy loss for every subsystem,
rj’. '"' - intemarioss factor of the subsystem,
ri.. - coupling loss factor between subsystems,
E- - the mean vibrational energy in Af frequency band in i-subsystem,
W - the input power carried into i-subsystem from outside.
To determine the elements of the coefficients matrix in eq. 1 it is necessary to
know [1,4,6] coupling loss factors (CLF) between structural subsystems and damping
In the SEA model in question transmission of the acoustic energy occurs in two
tvpes of connections between;
the plate and the beam (the point joint of the beam which is parallel to the edge
of the plate, (transfers flexural waves), . . „
two plates (linear connections and point joints transferring flexural waves).
The coupling loss factor between the plate and the beam which vibrate in the
flexural way (in the case of the point joint), is defined with following equation [1].
1.75c^
(2)
where:
c - is the speed of flexural wave,
T - transmission factor,
Q - the number of point connections,
Gj - angular frequency,
A - the surface of the plate.
1145
The speed of the flexural wave in the first plate c„ can be calculated in the
following way:
(3)
In the model under examination formula (2) defines coupling loss factors between the
anglesteel leg (2) and the beam (4) - ( factors 1,3. and tiJ or the beam (4) and the plate
(5) - rj45 and 7/54.
The flow of energy between two plates (which are connected at the right angle
and which vibrate in flexural way) is defined with the following formula in the case
of linear connection:
where:
1 - is the length of the connection.
With above formula it is possible to describe the flow of energy through correctly
made welded joints of plates or, for example, through bent plates. In the system
presented the factor determines the flow of energy between anglesteel legs (2) and
(3)
The coupling loss factor between two plates with a point joint is described by
the following formula:
iiL^
3 (o.X,
(5)
This type of connection occurs between the plate representing the anglesteel
leg and the plate (5). ^ .
After defining the value of factors matrix in the first equation it is possible to
specify the ratio of the vibroacoustical energy gathered in plate (1) and (5).
_ (n 2tof^ 3t0t~ ^ 24^ 32) 4fot^ 5tnt~ ^ 54^ 45^ ~ 34^ 43^ 5tot (5)
£5 ^21^ 32'^ 43^ 54
The damping loss factor is important parameter of every subsystem. For steel
plates used in the experiment the value of the damping loss factor have been measured
experimentally with the decay method. The results of the measurements have been
presented in figure 4. The frequency characteristic of rubber damping has been shown
in figure 5.
1146
DLF of steel [-]
frequency [Hz] frequency [Hz]
Fig. 4. Damping loss factor of steel Fig. 5. Damping loss factor of rubber
4. Experimental research
Experimental investigations have been carried out for connections made with
the use of rubber separator (elastic layer) of 50° Shore hardness. The connection was
build up of two identical, perpendicular plates connected each to another using the
anglesteel and the elastic rubber layer. These are the properties and material
parameters of individual elements:
plates:
- material constructional steel (St3);
- dimensions: 500 * 500 * 2.2 mm;
- Young modulus: 2,1 10'^ Pa.
anglesteel:
- material: constructional steel (St3);
- dimensions: L 40 40 2.2 mm;
- Young modulus: 2,1 10' ‘ Pa
elastic layer:
- material: rubber 50° Shore
- dimensions: 500 40 mm
- thickness: 2, 3, 4, 5, 6 mm
To avoid the loss of mechanical energy in the environment, during measure¬
ments the construction was suspended to the supporting frame with three weightless
strings in such way that only rigid body motions in the plane perpendicular to the plate
surface can occur. The excitation of the wide-band type with constant power spectral
density was applied in the symmetry axis of the plate (5) about 20 mm below the
upper edge (Figure 1). During the experiment the distribution of vibrating velocities
on the plate surface was obtain by non contact method using laser-vibrometry.
1147
5. FEM model
Vibration of modelled structure in low frequency has been analyzed by toe
Finite Element Method (FEM). Several FEM models, was build in order to consider
L valid thickness of the mbber layer. The mbber thickness has been from
0 mm to 6 mm. The dimensions and material parameters of the plates and the
LSesteel was constant. The structure was fixed in four comers. Calc^auons were
mfde for a division of the strucmre into 608 elements of type BRI^8. The mfluence
of the rubber thickness on the eigenfrequencies was remarkable. The results for first
15 eigenfrequencies are presented in the table 1 . In fig. was shown also the changes
of the value of natural frequencies in comparison with the natural frequencies of
strucmre without applying the mbber layer.
table 1. Namral frequencies of connection
LaDlc i. i>aLutai ai. — -
1 Natural frequencies (Hz) for various rubb
■ ■ . - - •
er layer tlrickness
0 nun
2 nun
3 mni
4 nun
5 nun
127.8
123.1
121.5
119.4
117.5
297.5
235.8
224.6
212.6
204.6
507.6
432.5
391.3
357.1
330.0
1003.2
566.9
525.7
503.7
491.9
1282.3
807.1
768.9
733.5
712.2
1900.5
1185.2
1169.4
1123.2
1070.2
2386.7
1570.2
1483.6
1376.2
1304.6
3272.3
1851.6
1634.2
1531.6
1490.8
3603.4
2303.1
2263.4
2167.8
2147.3
4263.3
2611.2
2539.5
2496.9
2463.7
4759.5
3192.7
3515
3110.1
2913.2
5438.4
3926.7
3874.6
3626.4
2937.0
7627.4
4110.1
4105.5
3639.9
3026.6
7691.6
5070.7
5137.0
3670.5
3111.2
11543.2
5547.1
5503.9
3721.9
3181.2
The decreasing of the absolute values of the natural frequences is observed according
to increasing of the thickness of the rubber layer. Beginning from die third of
°ib adons of the system the decrease of the natural frequencies is almost constant for the
"layer 2 mm or 3 mm and is continuously decreasing for rubber layer 4-6 mm (See
fig. 6).
1148
Fig. 6. Changes of natural frequencies
6. Comparison with experimental results
The equivalent coupling loss factor defining the energy flow between the plates has
been determined experimentally [31 for a model consisting of two subsystems; two plates.
The coupling loss factor in such two element model may be specified by the following
equation:
1 zast.
^2 ^1101
N, E2.0C
- 1
(7)
The quotient of plate energies E,/E, in a two-element model is relevant to the quotient of
energy in the first and fifth subsystem (E./E,) in the five element model presented m figure
0 the value of these quotient is defined with the equation (6)
In the picture we have presented the values of the equivalent coupling loss factor in
the connection. Individual points in the diagram show the results obtained experimentally.
The values received in computer simulation have been presented as a continuous diagram.
The upper curve shows the values of the coupling loss factor in the joint before the
application of the rubber layer.
In the frequency range above ca. 125 Hz we have received a very good comparison
of experimental results and computer simulation results performed with the AutoSEA
software At the frequency of about 200Hz there occurs a local minimum of the equivalent
coupling loss factor between the plates. The value of the minimum is essentially influenced
by the value of the rubber damping loss factor. The frequency (with the minimum CLF) is
strongly influenced by the peak frequency of the rubber damptng curve.
1149
o.ooo2ii -
symulation
experiment
welded plates
16' ‘ '40' ' ’lOO 250 630 1600 4000
frequency [Hz]
Fig.7 Equivalent CLF factor in the joint
7. Conclusion
A way of modelling the vibroacoustical energy flow with the help of SEA method has
been presented in the work. We have examined the screw connection of two plates, where
a rubber elastic layer has been applied, A comparison has also been made between the results
of computer simulation of the mechanical energy flow with SEA method and the experimental
results, and thus we have noticed the good correlation, especially as far as middle frequencies
The joint modification through introduction of the rubber separator has a remarkable
impact on the acoustic energy flow: , ^
- The application of the elastic layer in the Joint in question lowers the value ot
eauivalent coupling loss factor in the whole frequency range.
- the value of rubber damping factor has most significant influence on the acoustic
eneroy flow through connection in middle frequencies range:
“ The minimum value of the equivalent coupling loss factor in a joint is essentially
influenced by the rubber separator damping loss factor.
* The frequency of minimum CLF occurrence is strongly influenced by the peak
frequency of the rubber damping curve.
The increasing thickness of the rubber layer produces on decreasing natural frequencies of
the structure.
8. Bibliography:
III AutoSEA - User Guide Vibro-Acoustic Sciences Limited 1992.
121 Fahy F.J. Sound and Structural Vibration; Radiation, Transmission and Response Academic Pres
nTkirtuin J Smals N. Panuszka R. "Method of estimating the coupling loss factor for a set of
nlates" Mechanic, Technical University of Mining and Metallurgy, Cracow, 10,1991.
[^41 Lalor, N.: The evaluation of SEA Coupling Loss Factors. Proc. V School Energy Methods in
Vibroacoustics" - Supplement, Krakow-Zakopane 1996. , . „ „
[51 Lyon R., DeJong R.; Theory and Application of Statistical Energy Analysis. Butterworth-
Heinemann, Boston, 1995.
1150
Variation Analysis on Coupling Loss Factor
due to the Third Coupled Subsystem in
Statistical Energy Analysis
Hongbing Du Fook Fah Yap
School of Mechanical & Production Engineering
Nanyang Technological University
Singapore 639798
Abstract
Statistical Energy Analysis (SEA) is potentially a powerful method for
analyzing vibration problems of complex systems, especially at high frequen¬
cies. An impoitant parameter in SEA modeling is the coupling loss factor
which is usually obtained analytically based on a system with only two cou¬
pled elements. Whether the coupling loss factor obtained in the classical way
is applicable to a practical problem, which normally comprises of more than
two elements, is of importance to the success of SEA. In this paper, the varia¬
tion of coupling loss factor between two subsystems due to the presence of a
third coupled subsystem is investigated. It is shown that the degree to which
the coupling loss factor is affected depends on how strong the third subsystem
is coupled. It also depends on the distribution of the modes in the coupled sub¬
systems. This kind of effect will diminish when the damping is high, subsys¬
tems are reverberant, or ensemble-average is considered, but not for individual
cases.
1 Introduction
SEA is potentially a powerful method for analyzing vibration and acoustic problems
of complex systems, especially at high frequencies, because of the simplicity of
its equations compared to other deterministic analysis techniques. SEA models a
system in terms of interconnected subsystems. The coupling parameter between
any two subsystems is characterized by a coupling loss factor. If the coupling loss
factors and internal (damping) loss factors of all subsystems are known, the power
balance equation (e.g., see [1]) for each subsystem can be established. From this set
of equations, SEA predicts the system response (due to certain types of excitation)
in terms of the average energy of every subsystem. The energy can in turn be related
to other response quantities such as mean velocity or strain.
1151
Historically, the SEA power balance equations were initially derived from an
analysis of two coupled oscillators [2,3]. It has been shown that the energy flow
between them is directly proportional to the difference in their uncoupled modal
energies. The theory has then been extended to systems with multi-coupled sub¬
systems (e.g., [4]). Strictly this extension is only applicable if certain assumptions
are justified [4, 5]. Also the new concept of indirect coupling loss factor, which
is used to represent the energy flow proportionality between the indirectly coupled
subsystems, is also introduced.
In practice, the indirect coupling loss factors are normally ignored in SEA ap¬
plications because they are very difficult to determine analytically. Only coupling
loss factors between directly coupled substructures are considered. Some analy¬
ses [6-8] have shown that this approximation may lead to significant errors in the
predicted results if certain conditions are not met in the system. These conditions
include not only the well known requirement of weak coupling between subsystems
(e.g., see [6]), but also others, such as given by Langley that the response in each
element must be reverberant [7]; and by Kean that there should be no dominant
modes (peaks) inside the frequency-averaging band [8]. However, it is usually dif¬
ficult to know whether these conditions are satisfied for a particular system. In fact
the above mentioned conditions do not always hold for practical engineering cases.
On this point of view, the importance of a coupling loss factor for describing the
coupling between indirectly coupled subsystems are to be further examined.
A related question is whether the coupling loss factors obtained from the system
with only two subsystems can still be applied when other subsystems are present.
Generally, the coupling loss factor is sensitive to the amount of overlap between
the modes of the two coupled subsystems. When additional subsystems are cou¬
pled to the original two-subsystem model, the mode distributions of the originally
coupled two subsystems will be affected. The change of mode distributions will fur¬
ther affect the modal overlap between the coupled two subsystems and finally the
coupling loss factor between them. However, general estimation methods for cou¬
pling loss factor assume that the coupling parameters between two subsystems are
not affected much by the presence of the other subsystems. Therefore the conven¬
tional approaches of deriving coupling loss factor are mostly based on consideration
of a two-subsystem model only. One method is the wave approach, by which the
coupling loss factor used in the SEA applications are derived analytically from aver¬
aged transmission factors of waves that are transmitted through a Junction between
semi-infinite subsystems. This method only takes into account local properties at
the joints and sometimes may be inaccurate. Recent research [1,7,9-11] based
on the model with two-coupled subsystems has shown that the coupling parameter
does depend on other system properties, such as damping loss factor, etc. It can be
argued that, if there is a third coupled subsystem, the coupling parameters between
the first two subsystems will also depend on the energy flow to the third subsys¬
tem. Therefore, from a practical point of view, the coupling loss factor estimated
for two-coupled subsystems, ignoring the indirectly coupled subsystems, can only
be of approximate value.
1152
In this paper, the variation of coupling loss factor between two subsystems due
to the presence of a third coupled subsystem is studied. In the following sections,
the coupling loss factor is firstly expressed in terms of global mobility functions.
The exact solution of mobility functions is only for simple structures. However, for
general structures, it can be obtained by Finite Element Analysis (FEA) [10, 1 1].
The coupling loss factors obtained respectively in the cases with and without the
third subsystem in the model are compared for two particular system configura¬
tions, respectively. The system used in this investigation is one-dimensional simply
supported beanis coupled in series by rotational springs. By varying the spring stiff¬
ness, the strength of the coupling between the second and the third subsystems can
be changed. It is shown that the effect of the third coupled subsystem on the cou¬
pling loss factor between the first two coupled subsystems depends on how strong
the third subsystem is coupled. For each individual case, it is also shown that this
kind of effect may be positive or negative, depending on the distribution of modes
in the coupled subsystems.
2 Coupling Loss Factor by Global Modal Approach
In this section, a modal method is used to derive coupling loss factor in a sys¬
tem with any number of coupled subsystems. The result is then simplified for two
cases: (1) a three-subsystem model; (2) a two-subsystem model which is simply
substructured from the previous three-subsystem model by disconnected the third
subsystem.
For a linear system which consists of N coupled subsystems, if “rain-on-the-
roof” excitation [10] is assumed to be applied to each subsystem in turn, the corre¬
sponding response energy can be expressed as
I I drdsdu, (1)
sith.'ii/sl.Kmi Mii.bsy.'it.emj
where /?/; is the total time-averaged response energy of the subsystem i. due to
the excitation on the subsystem j, is the transfer mobility function be¬
tween the response points r and the excitation point H is the averaging range
of frequency, u.', in and S are the mass density and the power spectral density of
excitation. The input power due to the excitation is given by
I RelH(s,.i,L0)]d.sdu (2)
Q sahsyale'inj
where is the real part of the point mobility at the position
simplicity, two terms, a/, and are defined as
a,, =
i>, = ^ =
Al
m;S j
Ik
\ H [r, s,uj)\~ dr ds cl.in
n sii.bsy.'iUjvii /iubfiyslenij
lle[H{.^,s,ij)] dsd.u:
n subsyslr;mj
For
(3)
(4)
1153
The mobility function, H{r,.s,uj), is to be expressed in terms of the global modes
of the system, which can be obtained by Finite Element Analysis (FEA). By the
principle of reciprocity of the mobility function, the relation of aij = a ji always
holds regardless of the strength of coupling and the magnitude of input power if the
excitation is “rain-on-the-roof Theoretically applying the Power Injection Method
[12] we can obtain the SEA equation as
n = [77]E (5)
where H = {Hi, ila, - • • , and E = {E[, E-z, - - - , E^r}'^ . The SEA loss
factor matrix [77] is
iVl +
i.^2
(7?yV +
■niiau
niiCLiisr
bi
b-z
bN
■nua-ii
■nizCL-zN
bi
bo
bjV
77?.;Va,Yi
•m.A/a/VY
^1
Ih
b^!
= — B A“'M-
UJr
where, //; is the internal loss factor for subsystem i., v/,, is the coupling loss factor
from subsystem -I. to subsystem j, ujc is the central frequency of the averaging band
n.
A = [a,,]
From equation (6), the reciprocity principle of the coupling loss factors can be easily
seen, due to aij = aji.
1
''■bi
, M =
L ’-J
Vij n?. ; b j ni jbj/ (cu^/i / 2)
i]ji niibi ■niibi/iujc'^l'l)
Where, (77?.;6,:)/(u,v7r/2) is the averaged real part of point mobility [4] and can be
regarded as the generalized modal density of the subsystem i. Assuming weak
coupling and light damping, it approximately equals to the classical definition of
modal density [13]. Therefore, the relation given by equation (7) also reduces to the
classical reciprocity principle.
1154
2.1 Substructured two-subsystem model
Figure 1 : A general SEA model with three coupled subsystems in series
Consider a whole system with three substructures coupled in series as shown in
Figure 1 . If subsystem 3 is removed, the coupling loss factors between subsystems
1 and 2 are given in the equation
biCL22 bi(Li-2
"h '^712 — ^?2L _ ^ n?. i(fi 1 [fi22 — '^7.2(^11^22 — <'^i2^^'2l)
— 7/12 772+7/21 u-V _ ^20.21 _ _ ^2(1 1 1 _
777 i{an<^f22 “ <^^12f''2l) 7772(^110-22 — O12O21)
r bj ^to-12 1 (8)
^ J_ 777 ifli| 7772011022
62O21 ^2
7771O11O22 777.2 0 22
The approximation in the above equation is due to 011022 » 012O21 when the cou¬
pling is weak. Manipulating equation (8) with or without using the approximation
both can work out the coupling loss factors 7/12 and 7/21 as
(9)
r 61 6-,
( 777 1 I ) ( 77 72 62 ) - 7/2 - 77
07^-777-10 1 1 07^777.2022
! , ^2 . f. *^1
77 7 20'7 - 777 [Oi
"07c7772022 07^-777-1 011
(10)
The equations are true regardless of the strength of the coupling. It can be seen that
7/12 and 7/21 depend on the values of the three terms 777.,+/, 7// and bij{u:^ni-,au). The
first two are the generalized modal density and the internal loss factor, or in combi¬
nation equivalent to modal overlap factor. The third one, by noting the definitions
of 6; and an, is the ratio of input power to response energy for the directly excited
subsystem, i.e., the total loss factor of subsystem i. From equation (8), this term
can be approximately expressed as
Total loss factor
of subsystem i
b;
miLOcCLii
m + 5I'+-.7
(11)
1155
In the classical wave approach, where semi-infinite subsystems are assumed, the
total internal loss factors becomes
■n total, I = m + Vn and l]iot.al,2 = 112 + vTi (^2)
where is the classical coupling loss factor. Substituting equation (12) into equa¬
tions (9) and (10), i]ij reduces to the classical iiff, which only depends on the local
properties at the joints rather than other properties of the system, such as damping.
If the subsystem modal parameters are used to evaluate the term, bil{ujcmiaii), then
the total loss factor is Just the internal loss factor of the subsystem and the cou¬
pling loss factor is equal to zero. This is reasonable because using the uncoupled
modal parameters instead of the coupled modal parameters is actually equivalent to
removing the coupling between two subsystems.
However, for finite system where the assumption of semi-infinity is not justified,
there will be no immediate simplication for equations (9) and (10). Numerically,
FEA can be employed to obtain the global modes and then the coupling loss factor
can be calculated [10, 1 1].
2.2 Full three-subsystem model
Instead of substructuring, consider the three-subsystem model as a whole system,
shown in figure 1. Now the order of equation (6) is reduced to 3. With the global
modal parameters obtained from FEA, the coupling loss factors can be directly
evaluated. However, when the coupling between subsystems is weak, the order-
reduced equation (6) is still able to be simplified. Matrix A may be alternatively
expressed in the form of
■ a, L
0
0 ‘
■ 0
«L2
0
■ 0
0
«13 '
A -
0
(t-ll
0
-f
a\2
0
d.23
+
0
0
0
0
0
«33 .
0
((•23
0 .
. (l.[3
0
0
where the terms on the right side are sequentially defined as Ai, A2 and A3 . Under
the assumption of weak coupling, the non-zero entries in Ai, A2 and A3 will be of
the order O(t^), (9(e^) and respectively [14], The inverse of matrix A may
be approximately written as
= Ai”^ — Ai“^ A2 Ai”^ — Ai"^ A3 Ai ^ 4- Ai ^A2Ai ^A2Ai
+ •••
(14)
As an approximation, substituting only the first three terms in equation (14) into
equation (6) gives
[tj] ^ l/uv-
/>■
bo f t’ 12
rn I f( 1 1 a 2 2
b3(l-\3
ni[a\ KM. 3
■ni2a 1 1 (1-22
b2
1112(1.22
63^23
1112(1.22^(33
bi(t.\3
in3(l\\(l.33
bid. 13
in.3Ct.2'i((33
63 ~
1((3((33
(15)
1156
Generally, under the condition of weak coupling, the indirect coupling loss factors,
■ihs and 7731, are much smaller than the direct coupling loss factors and the internal
loss factors [5,7, 15]. The diagonal elements in [rj] can therefore be approximated to
the sum of internal loss factor and direct loss factor. It can be shown from equation
(15) that equation (11) remains valid for three coupled subsystems. But in the three-
subsystem case, the term, b,|{uJcm.^au), is to be evaluated by using the global modes
of the three-subsystem model.
3 Numerical Examples and Variation Analysis
In this section, two examples with different configurations are used to show the vari¬
ation of coupling loss factor due to the presence of a third coupled subsystem. The
coupling loss factor of the two-subsystem model is evaluated by using equations (9)
and (10). For the three-subsystem model, equations (6) and (15) are used. It can be
shown that both equations (6) and (15) give the same results as the couplings are
weak.
3.1 Structural details and SEA model
Ki
beam I (TJT] -
(a) ^ - Zi.
beam 1 rnn beam 2 (TTH ^
(b) ^ ^ - A A -
Figure 2; A structural model comprising of three beams
To begin example calculations, consider initially a two-subsystem model (figure 2(a))
which is two thin beams coupled through a rotational spring. The group of flexural
vibration modes of each beam are taken as a SEA subsystem. The spring provides
weak coupling between them where only rotational moment is transmitted. When
beam 3 is connected at the free end of beam 2 to the original two-beam model,
a three-subsystem model is formed (figure 2(b)). The specifications for the three
beams are given in table 1. The spring stiffness, A'2, is adjustable in order to look
into the significance of the effect of the third subsystem. There are two cases where
the length of beam 2 is: (i) L2 - 1.0?72; (ii) L2 = l-lm. The spring constants at
the joints are chosen to be weak enough to ensure that:(a) the coupling loss factor
is much smaller than the internal loss factor; (b) the indirect coupling loss factor is
much smaller than the direct coupling loss factor.
In the global modal approach (see section 2), the modes of two-subsystem model
and three-subsystem model are obtained from FEA. In numerical simulation, the
1157
Table 1 : The specifications of the three beams
BEAM
1
2
3
length (m)
2.0
1.0 & 1.1
0.7
width (mm)
4
Thickness (mm)
2
Density (Kg/m^)
7890
Young’s Modulus (N/m'^)
196E+9
Poisson Ratio
0.29
spring constant, A*i (Nm/rad)
1.0
central frequency is 200Hz and the averaging band is selected as 100 ~ 300Hz. In
order to take into account the contribution from the modes out of the band, all the
modes up to 500Hz are extracted for evaluating the mobility functions in averaging.
The modal loss factor is assumed to be the same for each modes used in averaging.
This means that the internal loss factor is the same for each subsystem and is equal
to the modal loss factor [11]. The results given are plotted against the modal loss
factor in order to show the damping effect at the same time.
3.2 Results and discussion
Figure 3 shows the identified coupling loss factor 7/12 for the case (i) {Lo = 1.0777)
with different stiffness of /v'2. The case of /v'2 = 0 means that the third subsystem
Figure 3: 7/12 is negatively affected in three-subsystem model
is not present. It can be seen that the coupling loss factor 7^12 is decreased in the low
1158
range of damping while the strength of the coupling between subsystem 2 and 3 is
increased. The stronger the coupling, the more ijn is decreased.
On the other hand, for the case (ii) where L2 = l.lm, the different results are
shown in figure 4 where the presence of the third subsystem would mainly increase
i]i2 in the low range of damping. The increasing magnitude is also dependent on the
strength of coupling between subsystem 2 and 3. The explanation for the different
variation trends of ?]i2 due to the third coupled subsystem between figure 3 and 4
will be given later.
Figure 4; 771 ■; is positively affected in three-subsystem model
From figure 3 and 4, the effect of damping on the coupling loss factor can also be
observed. In the low damping region, increasing damping would increase coupling
loss factor. After a certain turnover point, increasing damping would make the
coupling loss factor decrease and finally 7712 becomes convergent to a value. This
agrees with the conclusions drawn in [10, 11]. It is shown that, even though the
length of beam 2 has a slight difference in figure 3 and 4, the converged values are
still very close. Thus, the converged value seems not to depend on the variation of
coupling strength at A'2 and the structural details, although, with the third subsystem
existing in the system, the convergent speed is faster. Therefore, it is reasonable to
believe that the converged coupling loss factor at sufficiently high damping only
depends on the property of the joint rather than other system properties. This joint-
dependent property of coupling loss factor in the high range of damping accords
with the assumption in the wave approach. Here, it is convenient to define the
convergent region in the figure 3 and 4 as the “joint-dependent zone”.
However, before the “joint-dependent zone”, coupling loss factor seems very
sensitive to the variation of damping loss factor as well as the strength of coupling
between subsystem 2 and 3. It is because in the low damping region the system
1159
modal properties have been playing a major role in determining coupling loss fac¬
tor [10, 11]. In general, the coupling loss factor represents the ability of energy
transmitted between subsystems. It depends not only on the physical strength of
the coupling (e.g., spring stiffness in the examples), but also on the amount of over¬
lap between the modes of two connected subsystems. The higher modal overlap
between the modes of two connected subsystem, the more energy is transmitted
between the subsystems. As a result, the coupling loss factor will be higher even
though the physical strength at the joint is unchanged. If the modes in one subsys¬
tem are distributed exactly the same as those in one another(for instance, two exactly
same structures are coupled together), the coupling loss factor would be varied to
the maximum, and vice versa. Therefore, this region could be likely defined as
“modal-sensitive zone”.
In the “modal-sensitive-zone”, the dependence of coupling loss factor on the
amount of overlap between the modes of two connected subsystems has been clearly
shown in figures 3 and 4. For the case (i) illustrated in figure 3, the length of beam 2
is half of beam 1. Due to the characteristic of mode distribution in beam structure,
the amount of overlap between the modes of subsystem 1 and 2 is more than that in
the case (ii) shown in figure 4, where beam 1 is 2 meters and beam 2 is 1.1 meters.
Therefore, case (i) has higher coupling loss factor than case (ii) in “modal-sensitive
zone”. When the third beam is coupled, the induced variation of coupling loss factor
depends on how the amount of overlap between the modes of subsystems 1 and 2 is
affected. It can be increased or decreased and thus the coupling loss factor between
subsystems 1 and 2 can also be increased or decreased due to the third coupled
subsystem. For example, the amount of such overlap in case (i) is decreased after
the third subsystem is coupled. As a result, the coupling loss factor, 7712, becomes
decreased.
The above discussed variability of coupling loss factor due to the third coupled
subsystem has been shown for individual cases. On the other hand, if an ensemble
of similar structures are considered, this sensitivity may be reduced (as it is some¬
times positive or negative depending on each special situation). However, such a
variability obtained from two typical examples is nevertheless very useful when
one individual case is studied in SEA or SEA-like problems. The ignorance of such
effect of the other coupled subsystems on the coupling loss factor may become one
of the possible error sources causing SEA failure.
4 Conclusions
The variation of coupling loss factor due to the third coupled subsystem is stud¬
ied in this paper. The effect of a third coupled subsystem on the coupling loss
factor between the first two coupled subsystems depends on how strong the third
subsystem is coupled. Roughly, along with the damping in the subsystems, “joint-
dependent zone” and “modal-sensitive zone” are defined according to the different
variation properties of coupling loss factor. In the “modal-sensitive zone”, the ef-
1160
feet of a third coupled subsystem on the coupling loss factor could be positive or
negative. It depends on how the amount of overlap between the modes of two con¬
nected subsystems is affected. This “modal-sensitive” effect may be averaged out
for an ensemble of structures, but it is important when SEA is applied to individual
cases. In the “joint-dependent zone”, the coupling loss factor is insensitive to the
strength of the coupling between the second and third subsystems. Since the two
different zones are allocated according to the system damping (which is equivalent
to modal overlap factor when the central frequency and modal density are fixed),
it shows the importance of reverberance in subsystems when the classical SEA is
applied [7,8]. How to take into account the effect of the other coupled subsystems
in evaluating coupling loss factor, especially when the system damping is low and
when an individual case is considered, definitely needs to be further investigated.
References
[ 1 ] Richard H. Lyon and Richard G. DeJong. Theory and Application of Statistical
Energy Analysis. Butterworth-Heinemann, second edition, 1995.
[2] Richard H. Lyon and G. Maidanik. Power flow between linearly coupled os¬
cillators. Journal of the Acoustic Society of America, 34:623-639, 1962.
[3] Eric E. Ungar. Statistical energy analysis of vibrating systems. Transactions
of the ASME, Journal of Engineering for Industry, pages 626-632, November
1967.
[4] F. J. Fahy. Statistical energy analysis. In R. G. White and J. G. Walker, editors.
Noise and Vibration, chapter 7, pages 165-186. Chichester, Ellis Horwood,
1982.
[5] J. M. Cuschieri and J. C. Sun. Use of statistical energy analysis for rotating
machinery, part II; Coupling loss factors between indirectly coupled substruc¬
tures. Journal of Sound and Vibration, 170(2): 191-201, 1994.
[6] P. W. Smith. Statistical models of coupled dynamical systems and the transi¬
tion from weak to strong coupling. Journal of the Acoustic Society of America,
65:695-698, 1979.
[7] S. Finnveden. Ensemble averaged vibration energy flows in a three-element
structure. Journal of Sound and Vibration, 187(3);495-529, 1995.
[8] A. J. Keane. A note on modal summations and averaging methods as ap¬
plied to statistical energy analysis (SEA). Journal of Sound and Vibration,
164(1); 143-156, 1993.
[9] B. R. Mace. The statistical energy analysis of two continuous one-dimensional
subsystems. Journal of Sound and Vibration, 166(3):429-461, 1993.
1161
[ 10] Hongbing Du and Fook Fah Yap. A study of damping effects on coupling loss
factor used in statistical energy analysis. In Proceedings of the Fourth Inter¬
national Congress on Sound and Vibration, pages 265-272, St. Petersburge,
Russia, June 1996.
[11] Fook Fah Yap and J. Woodhouse. Investigation of damping effects on statis¬
tical energy analysis of coupled structures. Journal of Sound and Vibration,
197(3):35I-371, 1996.
[12] D. A. Bies and S. Hamid. In situ determination of loss and coupling loss
factors by the power injection method. Journal of Sound and Vibration,
70(2): 187-204, 1980.
[13] L. Cremer, M. Heckl, et al. Structure-Borne Sound: Structural Vibrations and
Sound Radiation at Audio Frequencies. Springer- Verlag, second edition, 1987.
[14] R. S. Langley. A derivation of the coupling loss factors used in statistical
energy analysis. Journal of Sound and Vibration, 141(2):207-219, 1990.
[15] J. C. Sun, C. Wang, et ai. Power flow between three series coupled oscillators.
Journal of Sound and Vibration, 1 89(2) :2 15-229, 1996.
1162
THE EFFECT OF CURVATURE UPON VIBRATIONAL
POWER TRANSMISSION IN BEAMS
SJ. Walsh(l) and R.G.White(2)
(1) Department of Aeronautical and Automotive
Engineering and Transport Studies
Loughborough University
(2) Department of Aeronautics and Astronautics
University of Southampton
ABSTRACT
Previous research into structural vibration transmission paths
has shown that it is possible to predict vibrational power
transmission in simple beam and plate structures. However, in
many practical structures transmission paths are composed of more
complex curved elements; therefore, there is a need to extend
vibrational power transmission analyses to this class of structure.
In this paper, expressions are derived which describe the vibrational
power transmission due to flexural, extensional and shear types of
travelling wave in a curved beam which has a constant radius of
curvature. By assuming sinusoidal wave motion, expressions are
developed which relate the time-averaged power transmission to
the travelling wave amplitudes. The results of numerical studies
are presented which show the effect upon power transmission
along a curved beam of: (i) the degree of curvature; and (ii) various
simplifying assumptions made about the beam deformation.
1. INTRODUCTION
Previous research into structural transmission paths has
shown it is possible to predict vibrational power transmission in
simple beam and plate structures. More recently, transmission
through pipes with bends, branches and discontinuities has been
studied, which has led to useful design rules concerning the
position and size of pipe supports for minimum power
transmission[l]. However, in many practical structures
transmission paths are composed of more complex curved
elements. Therefore, there is a need to extend power transmission
analyses to this class of structure.
1163
Wave motion in a curved beam with a constant radius of
curvature has been considered by Love [2] who assumed that the
centre-line remains unextended during flexural motion, whilst
flexural behaviour is ignored when considering extensional
motion. Using these assumptions the vibrational behaviour of
complete or incomplete rings has been considered by many
researchers who are interested in the low frequency behaviour of
arches and reinforcing rings. In reference [2] Love also presented
equations for thin shells which include the effects of extension of
the mid-surface during bending motion. Soedel [3] reduced these
equations and made them applicable to a curved beam of constant
radius of curvature. In an alternative approach Graff [4] derived
these equations from first principles and also constructed frequency
verses wavenumber and wavespeed versus wavenumber graphs.
Philipson [5] derived a set of equations of motion which included
extension of the central line in the flexural wave motion, and also
rotary inertia effects. In a development analogous to that of
Timoshenko for straight beams, Morley [6] introduced a correction
for radial shear when considering the vibration of curved beams.
Graff later presented frequency versus wave number and wave
speed versus wave number data for wave motion in a curved beam,
when higher order effects are included [7].
In this paper, expressions for vibrational power transmission
in a curved beam are derived from first principles. In the next
section two sets of governing equations for wave motion in a
curved beam are presented both of which include coupled
extensional-flexural motion. The first set is based upon a reduction
of Love's thin shell equations mentioned above. The second set is
based upon a reduction of Fliigge’s thin shell equations [8]. In
section three, the expressions for stresses and displacements
presented in section two are used to derive formulae for vibrational
power transmission in terms of centre-line displacements. By
assuming sinusoidal wave motion, expressions are developed
which relate the time-averaged power transmission to the
extensional and flexural travelling wave amplitudes. In section
four, corrections for rotary inertia and shear deformation are
introduced. The results of numerical studies of these expressions
are presented which show the effect upon wave motion and power
transmission of (i) the degree of curvature, and (ii) the various
simplifying assumptions made about the beam deformation.
1164
2. WAVE MOTION IN CURVED BEAMS
In this section the governing relations between displacements,
strains, stresses and force resultants in a curved beam are presented.
The centre-line of the beam lies in a plane and forms a constant
radius of curvature. The cross-section of the beam is uniform and
symmetrical about the plane and it is assumed that there is no
motion perpendicular to the plane. It is also assumed that the beam
material is linearly elastic, homogeneous, isotropic and continuous.
Consider a portion of the curved beam, as shown in Figure 1.
The circumferential coordinate measured around the centre-line is
s, while the outward pointing normal coordinate from the centre¬
line is z, and the general radial coordinate is r. A complete list of
notation is given in the appendix. For small displacements of thin
beams the assumptions, known as "Love's first approximation" in
classical shell theory, can be made [8]. This imposes the following
linear relationships between the tangential and radial
displacements of a material point and components of displacement
at the undeformed centre-line:
U (r, s, t) = u (R, s, t) + z (|) (s, t) (1)
W (r, s, t) = w (R, s, t) (2)
where u and w are the components of displacement at the centre¬
line in the tangential and radial directions, respectively, (j) is the
rotation of the normal to the centre-line during deformation:
(|>
3w
/ angle of'X { rotational displacement^
Vcurvature/ \ of straight beam )
(3)
and W is independent of z and is completely defined by the centre¬
line component w.
Circumferential strain consists of both an extensional strain
and bending strain component. Expressions for these are listed in
table 1. The strain-displacement expressions of the Love and Flugge
based equations are identical. However, in the total circumferential
strain of the Love based theory, the term in the denominator
has been neglected with respect to unity. Assuming the material to
be linearly elastic, the circumferential stress-strain relationship
is given by Hooke's Law, whilst the shear strain, Ysr/ and shear
stress, Osr/ are assumed to be zero. Assuming the material
1165
to be homogeneous and isotropic, the material properties E, G and v
can be treated as constants. Thus, by integrating the stresses over
the beam thickness, force and moment resultants can be obtained,
which are listed in table 2. The adopted sign convention is shown in
Figure 2.
Equations of motion for a curved beam are presented in [4].
These equations are derived in terms of the radian parameter 0. By
applying the substitution, s - RQ, the equations of motion can be
expressed in terms of the circumferential length, s. These equations
are listed in [9] along with the Fliigge based equations of motion
which have been obtained by a reduction of the equations of
motion for a circular cylindrical shell presented in [8]. An harmonic
solution of the equations of motion can be obtained by assuming
that extensional and flexural sinusoidal waves propagate in the
circumferential direction. The harmonic form of the equations of
motion are also listed in [9].
3. VIBRATIONAL POWER TRANSMISSION IN CURVED BEAMS
In this section the expressions for displacements and stresses
presented in section two are used to derive the structural intensity
and power transmission due to flexural and extensional travelling
waves in a curved beam. The structural intensity expressions are
formulated in terms of displacements at the centre-line. By
assuming sinusoidal wave motion, expressions are developed
which relate the time-averaged power transmission to the flexural
and extensional travelling wave amplitudes.
Structural Intensity in the circumferential direction of a curved
beam is given by [10]:
Is =
au
at
f intensity due to
I circumferential stress
aw
at
f intensity due to
radial shear stress
(4)
By integrating across the beam thickness power transmission per
unit length in the circumferential direction is obtained:
h/2
Ps= J Isdz (5)
-h/2
1166
Substituting the Love based circumferential stress-strain relation
and strain-displacement expression into equation (5) the power
transmission due to circumferential stress is obtained. (A full
derivation is given in [9].) By analogy to power transmission in a
straight beam [1] this can be expressed in terms of an extensional
component, Pe, and a bending moment component, Pbm* Although
the transverse shear stress Ogr is negligible under Love's first
approximation, the power transmission due to transverse shear
stress can be evaluated from the non-vanishing shear force, Q,
because the radial displacement W does not vary across the beam
thickness. Again, by analogy to power transmission in a straight
beam [1] this is expressed as a shear force component. Thus, the total
power transmission in the circumferential direction is given by the
sum of the extensional, bending moment and shear force
components. These equations are listed in table 3 along with Fliigge
based power transmission equations which are also listed in table 3.
Substituting harmonic wave expressions into the Love and Fliigge
based power transmission equations gives expressions for the power
transmission in the circumferential direction in terms of travelling
wave amplitudes A and B. For sinusoidal wave motion it is useful
to develop time-averaged power transmission defined by [1]:
T/2
<Ps>
-- f
t“T J
Ps (s,t) dt
- T/2
(6)
where T is the period of the signal. Time averaged Love and Fliigge
based power transmission equations are given in table 4.
4. THE EFFECT OF ROTARY INERTIA AND SHEAR
DEFORMATION
It is known that shear deformation and rotary inertia effects
become significant for straight beams as the wave length approaches
the same size as the thickness of the beam, and for cylindrical shells
as the shell radius decreases [8]. Thus, the objective in this section is
to establish more complete equations for power transmission in a
curved beam and to show under what conditions these specialise to
the simple bending equations presented in section three.
Rotary inertia effects are included by considering each element
of the beam to have rotary inertia in addition to translational
inertia. Equations of motion for a curved beam which include the
1167
effect of rotary inertia are presented in [7]. These equations are
listed in [9] in terms of the circumferential distance parameter, s.
Equations for vibrational power transmission can be derived in the
same marmer as described in section three. These equations are
listed in tables 3 and 4 where it can be seen that the extensional and
bending moment components when including rotary inertia effects
are identical to the corresponding Fliigge based expressions.
However, the shear force component now contains an additional
rotary inertia term.
If shear deformation is included then Kirchoff s hypothesis is
no longer valid, and the rotation of the normal to the centre-line
during bending, (|), is no longer defined by equation (3) but is now
another independent variable related to the shear angle, y.
However, unlike simple bending theory, where the transverse
shear strain, Ysr^ is negligible, the transverse shear strain is now
related to the shear angle, y which is expressed in terms of
displacements u, w and (j). The circumferential force, bending
moment, and shear force obtained from [8] are given in table 2. A set
of equations of motion for a curved beam which includes the effect
of shear deformation is presented in [9]. Power transmission
equations in the circumferential direction can be obtained in a
manner analogous to that used for Love and Fliigge based theories.
As before, the power transmission due to circumferential stress
can be identified as consisting of extensional and bending moment
components. The contribution to the power transmission from the
transverse shear stress is obtained from the product of the shear
force resultant and the radial velocity which gives the shear force
component of power transmission.
5. NUMERICAL STUDY
For a given real wavenumber, k, the harmonic equations of
motion were solved to find the corresponding circular frequency, co,
and complex wave amplitude ratio. The simulated beam was
chosen to have the physical dimensions and material properties of
typical mild steel beams used for laboratory experiments. Four
different radii of curvature were investigated, which were
represented in terms of the non-dimensional thickness to radius of
curvature ratio, h/R. These ratios were ^/lO/ 100/ 1000 and
^/lOOOO-
Using the Love equations of motion. Figure 3 shows the
relationship between wave number and frequency for a beam with a
1168
thickness to radius of curvature ratio of ^/lo- The frequency range
is represented in terms of the non-dimensional frequency
parameter Q = ®^/co, where Cq is the phase velocity of extensional
waves in a straight bar and the wave number range is represented
in terms of the non-dimensional wave number, kR. It can be seen
that two types of elastic wave exist: one involving predominantly
flexural motion; the other predominantly extensional motion.
However, for wave numbers less than kR = 1, the predominantly
flexural wave exhibits greater extensional than flexural motion.
Solution of the shear deformation equations of motion for a
curved beam shows that three types of elastic wave exist. These are
the predominantly flexural and predominantly extensional waves
of simple bending theory and additionally a predominantly
rotational wave related to the shear angle. The relationship
between wave number and frequency for these three wave types is
shown in Figure 4.
A numerical investigation of the power transmission
equations was undertaken using simulated beams with the same
dimensions and material properties as those used in the previous
study of wave motion. Figure 5 shows the relationship between
transmitted power ratio and frequency. For the predominantly
flexural wave the time-averaged transmitted power ratio is
calculated by dividing the time-averaged power transmitted along a
curved beam by a predominantly flexural wave by the time-
averaged power transmitted by a pure flexural wave travelling in a
straight Euler-Bernoulli beam. i.e. the ratio (<Pe>t + <Fbm>t +
<Psf>t)/EIcokf3Af . For the predominantly extensional wave the
transmitted power ratio is calculated by dividing the time-averaged
power transmitted along a curved beam by a predominantly
extensional wave by the time-averaged power transmitted by a pure
extensional wave in a straight rod. i.e. the ratio (<Pe>t + <Pbn>t +
<Psf >t)/EScokex P^ex
6. SUMMARY AND CONCLUSIONS
In this paper, starting from first principles, expressions for
vibrational power transmission in a curved beam have been
derived using four different theories. Love’s generalised shell
equations include extension of the centre-line during bending
motion were the first set of equations considered. Fiiigge's
equations also include centre-line extensions and were the second
set of equations used. Corrections for rotary inertia and shear
deformation produced the third and fourth sets of governing
1169
equations, respectively. By letting the radius of curvature, R, tend
to infinity these equations reduce to the corresponding straight
beam expressions presented in [1].
Using the governing equations for each theory, expressions
were then developed which related time-averaged power
transmission to the amplitudes of the extensional, flexural and
rotational displacements. For each theory the effects of curvature
upon the resulting wave motion and power transmission were
then investigated using beams with different degrees of curvature.
From the results of this study it can be seen that vibrational power
transmission in curved beams can be classified into three different
frequency regions:
(i) below the ring frequency, Q = 1, curvature effects are
important;
(ii) above the ring frequency but below the shear wave cut-on
frequency, = 1 the curved beam behaves essentially as a
straight beam;
(iii) above the shear wave cut-on frequency, higher order effects
are important.
ACKNOWLEDGEMENT
The analytical work presented in this paper was carried out while
both authors were at the Institute of Sound and Vibration Research,
University of Southampton. The financial support of the Marine
Technology Directorate Limited and the Science and Engineering
Research Council is gratefully acknowledged.
REFERENCES
1. J.L. HORNER 1990 PhD thesis, University of Southampton
Vibrational power transmission through beam like structures.
2. A.E.H. LOVE 1940 Dover, Nezv-York. A treatise on the
mathematical theory of elasticity.
3. W. SOEDEL 1985 Dekker, New York. Vibrations of shells and
plates
4. K.F. GRAFF 1975 Clarenden Press, Oxford. Wave motion in
elastic solids
5. L.L. PHILIPSON 1956 Journal of Applied Mechanics 23, 364-
366. On the role of extension in the flexural vibrations of rings.
6. L.S.D. MORLEY 1961 Quarterly Journal of Mechanics and
Applied Mathematics 14, (2), 155-172. Elastic waves in a
naturally curved rod.
1170
7. K.F. GRAFF 1970 IEEE Transactions on Sonics and Ultrasonics,
SU-17 (1), 1-6. Elastic wave propagation in a curved sonic
transmission line.
8. A.W. LEISSA 1977 NASA SP-288, Washington DC Vibrations
of shells.
9. S.J. WALSH 1996 PhD thesis, University of Southampton.
Vibrational power transmission in curved and stiffened
structures.
10. A.J. ROMANO, P.B. ABRAHAM, E.G. WILLIAMS 1990 Journal
of the Acoustical Society of America 87. A Poynting vector
formulation for thin shells and plates, its application to
structural intensity analysis and source localization. Part I:
Theory.
APPENDIX: NOTATION
A flexural wave amplitude
Af amplitude of a purely flexural wave
B extensional wave amplitude
Bex amplitude of a purely extensional wave
C rotation wave amplitude
E Young’s modulus
G shear modulus
I second moment of area of cross-section of beam
Ig structural intensity in circumferential direction
K radius of gyration
M bending moment on cross-section of beam
N circumferential force on cross-section of beam
P transmitted power
Q shear force on cross-section of beam
R radius of curvature
S cross-sectional area of beam
T period of wave
LF displacement in circumferential direction
W displacement in radial direction
Co wavespeed of extensional waves in a straight bar
Cs wavespeed of shear waves in a straight bar
ds length of elemental slice of curved beam
eg total circumferential strain
h thickness of beam
k wavenumber
kex wavenumber of a purely extensional wave
kf wavenumber of a purely flexural wave
r coordinate in radial direction
1171
s coordinate in circumferential direction
u displacement at centre-line in circumferential direction
w displacement at centre-line in radial direction
z coordinate of outward pointing normal
Q. non-dimensional frequency
Ps bending strain
Y shear angle
Ysr transverse shear strain
Ej. radial strain
£g circumferential strain
0A phase angle of flexural wave
0C phase angle of rotational wave
K Timoshenko shear coefficient
?iex wave length of extensional waves in a straight bar
V Poisson's ratio
Oj. radial stress
Gg circumferential stress
0SJ transverse shear stress
d change in slope of normal to centre-line
CO radian frequency
W(r,s,t)
1172
Figure 2: Sign convention and force resultants on an elemental
slice of curved beam
Figure 3: Wave number v. frequency relationship for a curved
beam predicted using Love theory
Transmitted power ratio (Curved beam)/(Slraight beam) ^ Non-dimensional wavenumber
Table 1: Displacement, strain-displacement and stress-strain equations for a curved beam.
1175
Table 2: Force resultants for a curved beam.
1176
Table 3: Power transmission for a curved beam.
1177
Table 4: Time-averaged power transmission by a single harmonic wave
1178
A Parameter-based Statistical Energy
Method for Mid-frequency
Vibration Transmission Analysis
Sungbae Choi, Graduate Student Research Assistant
Matthew P. Castanier, Assistant Research Scientist
Christophe Pierre, Associate Professor
Department of Mechanical Engineering and Applied Mechanics
The University of Michigan
Ann Arbor, MI 48109-2125, USA
Abstract
Vibration transmission between two multi-mode substructures con¬
nected by a spring is investigated. A classical Statistical Energy
Analysis (SEA) approach is reviewed, and it is seen that some typ¬
ical assumptions which are valid at high frequencies lose accuracy
in the mid-frequency range. One assumption considered here is
that of an identical probability density function (pdf) for each reso¬
nant frequency. This study proposes a Parameter-based Statistical
Energy Method (PSEM) which considers individual modal informa¬
tion. The results of PSEM have good agreement with those of a
Monte Carlo technique for an example system.
Nomenclature
E[ ]
expected value
power transmitted between substructure i and k
IIfc(u;) .
total power transmitted to substructure k
Pi
power input to substructure i
Vi,
coupling loss factor (CLF)
fh
modal driving force for mode j of substructure 1
UJ
frequency [rad/sec]
l.l.-.lj.Ol
subscripts for decoupled Bar 1
2,2r,2„02
subscripts for decoupled Bar 2
El, ^2
blocked energy
Eoi , Eo2
Young’s modulus
Pl^P2
density
mi, m2
mass per unit length
Mu M2
total mass
Cl,C2
viscous damping ratio
Ai, A2
cross-sectional area
0
rH
0
nominal length
1179
Ti,T2
disordered length
£1,^2
ratio of disorder to nominal length
k
coupling stiffness
coupling ratio
Xi,X2
position coordinate
ai,a2
point- coupling connection position
Wi, W2
deflection
modal amplitude
mode shape function
NuN2
number of resonant frequencies
resonant frequencies
lower limit of resonant frequencies
upper limit of resonant frequencies
(Tl, (72
standard deviation of disorder
1 Introduction
Vibration transmission analysis between connected substructures in the
mid-frequency range is often a daunting prospect. Since the analysis at
high frequencies requires greater model discretization, the size and com¬
putational cost of a full structure model (e.g., a Finite Element model)
can become prohibitive. Also, as the wavelengths approach the scale of
the structural variations, uncertainties (tolerances, defects, etc.) can sig¬
nificantly affect the dynamics of the structure. Starting at what may be
called the mid-frequency range, deterministic models fail to predict the
response of a representative structure with uncertainties.
Therefore, in the mid-frequency range, a statistical analysis of vibra¬
tion transmission may be more appropriate. This approach is taken in
the procedure known as Statistical Energy Analysis (SEA) [1]. In SEA,
a structure is divided into coupled substructures. It is assumed that each
substructure exhibits strong modal overlap which makes it difficult to dis¬
tinguish individual resonances. Therefore, the resonant frequencies are
treated as random variables, each with an identical, uniform probability
density function (pdf) in the frequency range of interest. This assumption
greatly simplifies the evaluation of the expected value of transmitted vi¬
bration energy. A simple linear relation of vibration transmission between
each pair of substructures is retrieved. The power transmitted is propor¬
tional to the difference in the average modal energies of the substructures.
This relation is analogous to Fourier’s law of heat transfer [1-4].
In the low- to mid-frequency region, the modal responses are not strongly
overlapped. In this case, two typical SEA assumptions are less accurate: an
identical pdf for all resonant frequencies, and identical (ensemble-averaged)
1180
values of the associated mode shape functions at connection positions. In
this paper, these two assumptions are relaxed. A distinct uniform pdf
is applied for each resonant frequency, and a piecewise evaluation of the
transmitted power is performed. This is called a Parameter-based Sta¬
tistical Energy Method (PSEM) because it considers the statistical char¬
acteristics of individual system parameters. This solution can accurately
capture peaks of transmitted power while maintaining the SEA advantage
of efficiency.
This paper is organized as follows. In section 2, we briefly review SEA
along with the associated assumptions and limitations. In section 3, the
power transmitted between two spring- coupled multi-mode substructures
is investigated by applying several SEA assumptions. A Monte Carlo solu¬
tion is used for comparison. In section 4, the PSEM approach is presented
and the results are shown. Finally, section 5 draws conclusions from this
study.
2 Overview of SEA
In Statistical Energy Analysis, the primary variable is the time-averaged
total energy of each substructure. This is called the blocked energy, where
blocked means an assumed coupling condition. The assumed coupling con¬
dition may be the actual coupling, a clamped condition at the substructure
junctions, or a decoupled condition [2, 5].
In order to predict the average power transmitted between two directly-
coupled substructures, a few simplifying assumptions are applied. Some of
the essential SEA assumptions are summarized by Hodges and Woodhouse
in Ref. [3];
• Modal incoherence: the responses of two different modal coordinates
are uncorrelated over a long time interval
• Equipartition of modal energy: all modes within the system have the
same kinetic energy
The above conditions make it possible to treat all modal responses as sta¬
tistically identical. The first assumption implies a broad band, distributed
driving force (often called ”rain on the roof’) which leads to uncorrelated
modal driving forces. The second assumption implies that the substruc¬
tures have strong modal overlap, or that the parameter uncertainties are
sufficiently large that the modes are equally excited in an ensemble average
sense. Thus, the resonant frequencies are treated as random variables with
identical, uniform probability density functions (pdfs) for the frequency
range of interest.
The SEA relation for the expected value of power transmitted from
1181
substructure i to substructure k, may be expressed as
E[nijfc(cj)] = 0)7}, ^Ni{^ (1)
where uj is the frequency, 77^^ is the coupling loss factor, Ei is the blocked
energy of substructure z, and Ni is the number of participating modes of
substructure i for the frequency range of interest. The power dissipated by
substructure i is expressed as
= u:riiEi (2)
where r]i is the damping loss factor. Using Eqs.(l) and (2), the equation
of power balance for substructure i at steady state [1, 2, 5, 6] is
N N
Pi = E[Ilij] + UJTJiEi = UJ (3)
i=i
j¥i
where Pi is the power input to substructure i from external sources. Note
that the first term on the right-hand side is the power transmitted through
direct coupling between substructures.
3 Vibration Transmission in a Two-Bar
System
The longitudinal vibration of the structure shown in Fig.l is considered
in this study. The structure consists of two uniform bars with viscous
damping which are coupled by a linear spring of stiffness k. The spring is
connected at intermediate points on the bars, Xi = ai and X2 = 0,2. Bar
i has nominal length Loi. A parameter uncertainty may be introduced by
allowing the length to vary by a small random factor e^, which is referred
to as disorder. The length of a disordered bar is Li = Zrox(l + £:)• The ratio
of the connection position to the length, aifLi^ is held constant. Bar 1 is
excited by a distributed force Fi{xi,t).
3.1 Nominal transmitted power
The power transmitted from Bar 1 to Bar 2 for the nominal system (no
disorder) is briefly presented here. A more detailed derivation is shown in
the Appendix (see also Refs. [7] and [8]). The equations of motion are
Wi(a:i,t) -f ^Ci(wi(a;i,f)) = Fi(a:i,t)-|-
^[W2(a2, t)-wi(ai, - ci)
(jBo2A2-§^ + m2|^^W2(x2,t) + ^C'2(w2(a;2,t)) =
k[wi{ai,t)-W2{a2,t)]6{x2 - ^2)
1182
a)
Bar 2
Fig. 1: Two-bar system
where ^ is a Dirac delta function, and (for Bar i) Eoi is Young’s modulus,
Ai is the cross-sectional area, rm is the mass per unit length, ’Ci is the
viscous damping operator, and Wt(a:i,t) is the deflection. The deflections
of the two bars can be expressed by a summation of modes:
Wi{xi,t) = W2(x2,t) = E 1^2r(i)^2r(^2) (5)
t=0 r=0
where and W2^{t) are modal amplitudes, and and ^2r(^2)
are mode shape functions of the decoupled bars. These mode shape func¬
tions are normalized so that each modal mass is equal to the total mass of
the bar, Mi. Applying modal analysis and taking a Fourier transform, the
following equations are obtained:
Mi(f)ijWij = 4- fc^ij(ai)[E kF2r^2r(<^2) - E
_ (6)
M2<f>2,W2, = ^^^2,(<22)[E VFi,^i,(ai) - E W2.^2M2)]
i=0 r=0
[ul. - -f • 2Cia;i .a;)(2 - sgn(j))
y/^ • 2C2<^2.^)(2 - sgn(s))
1 for ^ > 0
0 for 2 = 0
where an over-bar (") denotes a Fourier transform, and are resonant
frequencies, (i and (2 are damping ratios, and is a modal driving force.
Mode 0 is a rigid body mode, which is why the sgn(i) term is present.
Note that the damping ratio of each bar is assumed to be the same for all
modes.
Next, the modal driving forces are assumed to be incoherent, and each
spectral density function is assumed to be constant (white noise) over a
finite range of frequency [2]. After some algebra, the power transmitted
<^2^ =
sgn(0 =
1183
(7)
from Bar 1 to Bar 2, 1112(0;), is found as:
2o;^PC2«^PlPl ^ ^2r(^2)0^2r
M1W2IAI2 ^ |(^2.|2
-^1 -^2 <?^2r
where 5pjpi is the same uniform spectral density function for each modal
driving force on Bar 1.
1112(0;) =
A =
3.2 Monte Carlo Energy Method (MCEM)
The disordered case is now considered, where each bar has a random length.
The ensemble-averaged transmitted power for a population of disordered
two-bar systems is found by tahing the expected value of Eq. (7):
^[ni2]
(8)
where Ni is the number of modes taken for Bar i (this is an arbitrary set of
modes that have been aliased to the numbers 1,2,- • ’,Ni). Since a truncated
set of modes is used, Eq. (8) is an approximation. The random variables
in Eq. (8) are the resonant frequencies of the bars (which are present in
the terms ^i;, ^2r? s,nd A).
Equation (8) may be solved numerically using a Monte Carlo method:
the random variables are assigned with a pseudo-random number generator
for each realization of a disordered system, and the transmitted power
is averaged for many realizations. This is called a Monte Carlo Energy
Method (MCEM) here. It may be used as a benchmark for comparing the
accuracy of other approximate methods.
Note that the resonant frequencies of a bar may be found directly from
the disordered length. Therefore, for the MCEM results in this study, the
actual number of random variables in Eq. (8) is taken to be one for each
decoupled bar. That is, the two random lengths are assigned for each real¬
ization, and then the natural frequencies are found for each bar in order to
calculate the transmitted power. If such a relation were not known, each
resonant frequency could be treated as an independent random variable.
3.3 SEA-equivalent Transmitted Power
An SEA approximation of the transmitted power may be obtained by ap¬
plying several typical SEA assumptions to Eq. (8). (Since Eq. (1) is not
used directly, this might be called an SEA-equivalent transmitted power.)
These assumptions were summarized in Ref. [8]: the coupling between
1184
substructures is weak, the modal responses are uncorrelated, the expected
value of the square of mode shape functions at connection positions is unity,
and the pdfs of the resonant frequencies are uniform and identical.
The assumption of weak coupling means that the value of jAj in Eq.
(8) is approximately one. Applying the second and third assumptions then
yields
E[ai2]
M?M2
(9)
Since the pdf of each resonant frequency is assumed to be uniform, the
expected values in Eq. (9) are
E[
E[
l/4u)"
1
.2cos|
-f 2a;a;i. • cosf +
^ ul. — 2u}ujii • cos| •
• sin I
sin%
ian~
U)
2 _
(10)
l/4a;"C2Vr^, - 1)
- 1 - tan — — - - —
W2r=‘*'2ru
— ^2r
4w^C2\/i - Cl
(11)
W2r=‘*'2rj
where a = cos~^(l - 2Cj), subject to the restrictions (1 - 2(1)^ < 1 and
(1 _ 2(2)2 ^ I Finally, since the pdfs of the resonant frequencies are taken
to be identical, the frequency limits do not depend on the individual modes
(a;i. = uj2r, - a^nd = u;2,„ = oju)- Therefore, each sum in Eq. (9)
simplifies to the product of the expected value and the number of modes
in the frequency range of interest:
E[n^2]
2C2A:ViViiV25,^p^ r.r 1 1 Pr ^2. ^
WM2 Vi.r
(12)
Equation (12) is the SEA approximation used in this study.
3.4 Example
The three formulations of the transmitted power presented thus far the
nominal transmitted power in Eq. (7), the MCEM transmitted power in
Eq. (8), and the SEA-equivalent transmitted power in Eq. (12) — are
now compared for a two-bar system with the parameters shown in Table 1.
For the MCEM results, the disorder (ei and 62) was taken to be uniformly
distributed with mean zero and standard deviation ai = 0-2 = 10%.
As a measure of the coupling strength, the coupling ratio, Ri, is defined
cLS the ratio of coupling stiffness to the equivalent stiffness of a bar at the
1185
Table 1: Material properties and dimensions of two bars
MIIM2
21.53/21,53
[Kg]
R01IL02
10.58/8.817
[m]
E01IR02
200 XIOV2OO xlO^
[N/m'1
ai/a2
2.116/7.053
N
Pllp2
7,800/7,800
[Kg/m'*l
^pipi
1
[N^l
C1/C2
0.005/0.005
k
4.868 xlO^
[N/m;
Fig. 2: Comparison of the nominal transmitted power, the MCEM results
(20,000 realizations with (j^ = <72- 10%), and the SEA approximation.
fundamental resonant frequency, Ri = weak cou¬
pling is considered here such that Ri = 0.01.
The nominal transmitted power, the MCEM results, and the SEA ap¬
proximation are shown in Fig.2. The transmitted power calculated for
the nominal system exhibits distinct resonances. This is due to the low
modal overlap of the bars in this frequency range. The MCEM results
show distinct peaks for uj < 15,000 rad/s, but they become smooth as
the frequency increases. The SEA approximation does not capture indi¬
vidual resonances. However, at the higher frequencies where the disorder
effects are stronger, the SEA approximation agrees well with the MCEM
results. The frequency range between where the MCEM results are close
to the nominal results and where they are close to the SEA results (ap¬
proximately 2,500 ~ 15,000 rad/s for this case) is considered to be the
mid-frequency range here. This range will vary depending on the system
1186
parameters and the disorder strength. In the next section, an efficient ap¬
proximation of the transmitted power is presented which compares well
with MCEM in the mid-frequency range.
4 Parameter-based Statistical Energy
Method (PSEM)
The SEA approximation presented in the previous section does not capture
the resonances in the transmitted power because of two assumptions: the
resonant frequencies all have the same uniform pdf, and the values of the
square of mode shape functions at the connection positions are taken to be
the ensemble- averaged value. Keane proposed an alternate pdf of resonant
frequencies in order to apply SEA to the case of two coupled nearly periodic
structures [9]. This pdf is shown in Fig.3(a). It accounts for the fact that
Fig. 3: (a) The pdf of the natural frequencies and the resultant transmitted
power from Ref. [9]. (b) The pdfs of three natural frequencies, and a
schematic representation of the piecewise evaluation of transmitted power
for PSEM. The individual modal contributions are extrapolated ( - ) and
summed to calculate the total transmitted power ( — ).
the natural frequencies of a nearly periodic structure tend to be grouped in
several distinct frequency bands. Thus the pdf has a large constant value
for those frequency bands, and a small constant value elsewhere. The SEA
approximation of transmitted power is then modified by simply adding a
positive value or negative value on a logarithmic scale, as demonstrated in
Fig, 3(a). This solution thus captures some of the resonant behavior of the
transmitted power.
1187
Here, a more general approach is taken for approximating the power
transmitted between two substructures in a frequency range in which they
have low or intermediate modal overlap. Each resonant frequency is as¬
signed a uniform pdf. However, the frequency range of each pdf is diiferent;
it corresponds to the range in which that resonant frequency is most likely
to be found. (The concept of using “confidence bands” as one-dimensional
pdfs was suggested but not pursued in Ref. [8].) An example is shown in
Fig. 3(b) for three resonant frequencies. Furthermore, it is assumed that
the values of the square of the mode shape functions at the connection
positions are known. Thus, applying only the first two SEA assumptions
along with those noted above, Eq. (8) becomes:
N2
i:kme
This is called a Parameter-ba.sed Statistical Energy Method (PSEM) be¬
cause it employs information for individual modal parameters.
Since each modal pdf is uniform, Eqs. (10) and (11) still hold for the
expected values in Eq. (13). However, unlike the SEA approximation, each
expected value is different, because the corresponding frequency bounds are
unique. Furthermore, note that the pdfs do not cover the entire frequency
range of interest. The results for each mode are therefore extrapolated
outside the frequency range of that modal pdf before the individual modal
contributions are summed. This is shown schematically in Fig. 3(b). PSEM
is therefore a piecewise evaluation of the expected value of transmitted
power.
The PSEM approximation is now applied to the two-bar system of
Table 1, with the standard deviation of disorder cti = 0-2 = 10%. The pdfs
of the resonant frequencies of Bar 1 and Bar 2 aze shown in Fig. 4(a) and
(b), respectively. For this system, the bounds for each resonant frequency
may be found directly from the variation of the uncertain parameter. It
can be seen that the spread of each natural frequency pdf due to disorder
increases with increasing frequency.
The MCEM, PSEM, and SEA approximations for the transmitted power
are shown in Fig. 4(c). There were 20,000 realizations taken for the MCEM
results at each sampled frequency. This took about 10 hours of computa¬
tion time. In contrast, the PSEM results only required 3 seconds of compu¬
tation time, and the SEA results only required about 1 second. Note that
the PSEM results show excellent agreement with the much more expensive
MCEM results. The difference at very low frequencies comes from the fact
that for the PSEM approximation, the value of the term |A| was assumed
to be one due to weak coupling. This assumption breaks down as the
frequency approaches zero. However, the match between the MCEM and
1188
Fig. 4: (a) Natural frequency pdfs for Bar 1. (b) Natural frequency pdfs
for Bar 2. (c) Transmitted powers obtained by MCEM, PSEM, and SEA
for c7i = <J2 = 10%.
PSEM results in the mid-frequency range is excellent. Again, it is noted
that the SEA results converge to those of MCEM (and PSEM) as the fre¬
quency increases. Now it can be seen that the assumption of identical pdfs
for all modes becomes better with increasing frequency.
Next, the example system is considered with smaller disorder, Ci =
(72 = 1%. Fig. 5 shows the results for this case for what might be called
the mid-frequency range. Note that even though this is a higher frequency
range than that considered for the previous ca.se, the pdfs of the resonant
frequencies shown in Fig. 5(a) and (b) are not as strongly overlapped.
Thus, several peaks are seen in the transmitted power in Fig. 5(c). Again,
the PSEM approximation agrees well with the MCEM results, although
there is more discrepancy for this case. The SEA approximation follows
the global trend, but does not capture the resonances or anti-resonances.
The SEA results drop off at the edges because only modes within this
frequency range are considered to contribute to the transmitted power.
In addition to PSEM, another piecewise evaluation of the transmitted
power is considered here. For this approximation, wherever the individual
mode pdfs overlap, they are superposed to form a pdf for all the modes in
that “section” of the frequency range. This superposition is demonstrated
in Fig. 6. Also, if the number of modes in a section is above a certain cutoff
number, Nc, then it is assumed that their mode shape function values at
the connection positions are unknown, so that the ensemble-averaged value
must be used. This is called a multiple mode approximation. The purpose
1189
Fig. 5: (a) Natural frequency pdfs for Bar 1. (b) Natural frequency pdfs
for Bar 2. (c) Transmitted power obtained by MCEM, PSEM,-and SEA
for cTi = 0-2 = 1%.
Fig. 6: Resonant frequency pdfs for PSEM and for the multiple mode
approximation.
of formulating this approach is to investigate what happens as information
about the individual modes is lost.
The multiple mode approximation is applied to the example system
with cTi = <72 = 10% in Fig. 7. For A/c = 2, this approximation has good
agreement with MCEM. The match is especially good for u) < 10,000.
Above this frequency, the number of overlapped resonant frequencies in
each pdf section is greater than Nc, and the loss of mode shape information
affects the results slightly. For Nc = 0, the values of the mode shape
functions are taken to be one for the entire frequency range, just as in the
SEA approximation. As can be seen in Fig. 7, the piecewise construction
of the pdf roughly captures the frequency ranges of the resonances and
anti-resonances. However, the mode shape effect is more pronounced in
1190
-B.S
- MCEM
-e.s — Multiple mode approximation rvt= 2
. Multiple mode approximation Nfc= O
Fig. 7: Transmitted power obtained from MCEM and the multiple mode
approximation for Ui = <72 = 10%
the mid-frequency range. The peak values are now similar to the SEA
approximation.
5 Conclusions
In this study, the power transmitted between two multi-mode substruc¬
tures coupled by a spring was considered. A Monte Carlo Energy Method
(MCEM) was used to calculate the ensemble average of the transmitted
power for the system with parameter uncertainties. A classical Statistical
Energy Analysis (SEA) approximation matched the Monte Carlo results
in the high-frequency range, but did not capture the resonant behavior of
the transmitted power in the mid-frequency range where the substructures
have weak modal overlap.
A Parameter-based Statistical Energy Method (PSEM) was presented
which uses a distinct pdf for each natural frequency as well as some indi¬
vidual mode shape information. A piecewise evaluation of the transmitted
power was performed, and then the modal contributions were extrapolated
and superposed. The PSEM approximation compared very well with the
much more expensive Monte Carlo results, including in the mid-frequency
1191
range.
References
1 R. H. Lyon. Statistical Energy Analysis of Dynamical Systems: Theory
and Applications. M.I.T. Press, 1st edition, 1975.
2 R. H. Lyon. Theory and Application of Statistical Energy Analysis.
Butterworth-Heinemann, 2nd edition, 1995.
3 C. H. Hodges and J. Woodhouse. Theories of noise and vibration trans¬
mission in complex structures. Rep. Prog. Physics, 49:107-170, 1986.
4 J. Woodhouse. An approach to the theoretical background- of statis¬
tical energy analysis applied to structural vibration. Journal of the
Acoustical Society of America, 69(6):1695-1709, 1981.
5 M. P. Norton. Fundamentals of Noise and Vibration Analysis for En¬
gineers. Cambridge University Press, 1st edition, 1989.
6 N. Lalor. Statistical energy analysis and its use as an nvh analysis tool.
Sound and Vibration, 30(l):16-20, 1996.
7 Huw G. Davies. Power flow between two coupled beams. Journal of
the Acoustical Society of America, 51(1):393-401, 1972.
8 A. J. Keane and W. G. Price. Statistical energy analysis of strongly
coupled systems. Journal of Sound and Vibration, 117(2):363-386,
1987.
9 A. J. Keane. Statistical Energy Analysis of Engineering Structures
(Ph.D Dissertation). Brunei University (England), 1988.
Appendix
In this appendix, the nominal transmitted power in Eq. (7) for the mono-
coupled two-bar system is derived. The procedure follows that of Refs [7,
8].
Plugging Eq. (5) into Eq. (4),
F^ixut) + fcf E - Oi) (A.1)
t=0
1192
k £ Wi,(<)$i,(a,) - £ lV2,(«)«2,(a2) 5(x2 - aj)
Multiplying Eq. (A.l) by and integrating with respect to xi for
[0, Li] yields
MiK + =
A, + f f; W2,(4)W2Xo2) - E
Li
2CijUijMi = y C'i(«'i,.(a:i))Wij(xi)cia:i , fi, J Fi{xi,t)^i.{xi)dx],
0 0
and wij is the ;th resonant frequency of decoupled Bar 1. The damping
ratio in Eq. (A. 3) is now assumed to be the same (Ci) for all modes, since
the differences in the ratio are usually small and this simplifies the equation.
Taking the Fourier transform of Eq. (A.3) with zero initial conditions leads
to the following
= 7i, + S:$i,(ai)fETr2, $2,(02)
t r=0 i=0 -*
</>!;• = K, - • 2(iu;i^iv){2 - sgn(i))
J fori = 0
where (~) denotes a Fourier transform. Similarly, applying the previous
procedure to Eq. (A.2),
r OO CO -1
M2(^2.1V2. = ^^$2.(02) EW"li®l.(<»0-EW"2,$2,(o2) (A.5)
t i=0 r=0
^2s = {^2, - • 2C2^^2,w)(2 - sgn(s)).
Solving for VF2, from Eq. (A.4) and (A.5),
= Trr-Wx-l^^- - (a.6)
where
_ W2.(a2).^/ii$i,(ai)
1193
Calculating the second term in brackets in Eq. (A. 6),
fc4'2.(<22)EW^2,«'2.(a2) E
where
Plugging Eq. (A. 7) into Eq. (A. 6),
M2<j>2s 1 + CKl + Q;2 1=0
\ I ' I. -I— ✓
Coupling force Ps
Using the definition of transmitted power in Ref. [7], 1112(0)) is
ni2 = Re[-v^o)f;E[P.F;,(o))]
L „_ri
"I" 0^1 + P i~0 j=:0 J
A
i<?^2.p hh Ml,
where Re[] denotes the real part of the argument, * is a complex conjugate,
and
Finally, it is assumed that the modal driving forces, f-^., are uncorrelated.
Also, the spectral density function of each modal driving force is assumed
to be constant for the finite frequency range of interest:
C ( \ f ^PlVl ^
V for i^j.
Therefore, Eq. (A. 9) becomes
2^2fc^O)^.5^pipi ^ ^ ^2r(^2)o)2r
to to
(A.IO)
(A.ll)
1194
PASSIVE AND ACTIVE CONTROL III
Research on Control Law of Active Siispension of a
Seven Degree of Freedom Vehicle Model
Dr&Prof. YuchengLei Lifen Chen
Automobile Engineering Dept, Tong Ji University ,Shang
Hai,P.R.of China
Abstract
In the paper , control law of active suspension is presented ,
which involves 7-DOF vehicle model for improving control
accuracy .The control law involve vehicle running velocity ,
road power spectrum , suspension stiffness and damping .The
control law can be applied to multi-DOF control of active
suspension of vehicle .
Keywords: Active suspension , control law , Game theory ,
Modeling , 7 — DOF Vehicle Model .
1 . Introduction
An individual control system for each wheel by applying the
optimum regulation method for the two degrees of freedom is
showed in [3] . [4] and [5] also introduce two-DOF feedback
control method of active suspension .It is difficult for two-
DOF control method to coordinate multi-DOF kinematic
distances of entire car . Muti-DOF active control can improve
coordination control accuracy of entire car , but high speed of
CPU is asked for control and calculation while control law of
multi-DOF is got by real-time calculation . And, ride
performance and handling performance is inconsistent . For
resolving the problem ,the paper holds a new calculation
method for optimizing the law that can be programmed for
real-time control by table-lookup and not by real-time
calculation .So the method and law can not only improve
coordination control accuracy ,but also develop control
speed .
2 . Mathematical Model
Vehicle is simplified to turn into 7 DOF model . 7-DOF
vibration motion equation can be written as follow
[Af ]z+ [c]z+ [kY = \C, ](2+ ]e (1)
1195
Where [M] is mass matrix , [C] is suspension damping
matrix , [A’] is suspension stiffness matrix , [C,] is tyre
• •
damping matrix , \K, ] is tyre stiffness matrix , Z is
acceleration matrix ,Z is velocity matrix , Z is 7-DOF
displacement matrix , Q is road surface input velocity matrix ,
Q is road surface input displacement matrix .
z=p„z....,zy (2)
•• . • -j-J
Where Zi is vehicle vertical acceleration, Zi is roll
•• ••
acceleration , Z3 is pitch acceleration , Z,- (/ = 4,*«*,7) is
four tyres vertical acceleration .
3 General Optimization Method of Control Law
Objective function of optimization of control law can
generally got by calculating weighted sum of 7-DOF mean
square root of acceleration , dynamic deflection and dynamic
load .it can be written as follows
/=! >1 A=1
Where a-^ (z = !,• • (7 = !,• • •A), r * = h* • •A) is
weighted ratio . Where a.. (i=l, * * ' ,7) is 7-DOF mean
Zi
square root of acceleration , o-jy_ 0=1? ‘ ' is 7-DOF mean
square root of dynamic deflection , (k=l, * * ^ A) is
7-DOF mean square root of relative dynamic load .cr.. ,
Zi
can be calculated by resolving (1) using numerical
method .
4 Result of General Optimization
Method of Control Law
Optimization result of control law of a truck is got using
above method as figure 1 and 2 , its main parameters as
follows .
Wheel distance is 1.4 meter , axle distance is 2.297 meter ,
mass is 1121.3 kg , front tyre and axle’ s mass is 22.8 kg ,
rear tyre and axle’ s mass is 35.0 kg , X axis’ rotational
1196
inertia is 307.4 kg-ni^ , Y axis’ rotational inertia is 1276.5
kg-rri^ .
In fig. 1 and 2 , Cl of RMSMIN and C3 of RMSMIN are
respectively front and rear suspension damping of getting
minimization of above objective function , it is changing
while road surface rough coefficient and automobile
velocity V is changing . Cl and C3 also rise when velocity V
rises . This is called control law of general optimization
method of active suspension in the paper . The result in fig. 1
and fig. 2 has been verified by road test .
Fig. 1 front suspension Fig. 2 rear suspension
optimization damping optimization damping
Simulation result can also verify that ride performance’s
increasing (suspension stiffness reducing) will make handling
performace reduce . So selecting perfect
= is very difficult
and inconsistent .The paper advances next game method to
try to resolve the inconsistent problem .
5 Game Optimization Method of Control Law
Because to select weighted ratio of general optimization
method is difficult , the paper advances a new method of
optimization of control law — Game Balance Optimization
Method 4t is discussed as follows .
Game theory method of two countermeasure aspect can be
expressed as follows :
1197
— 1 j ^ 0 ,z •— 1, 2,
tr (4)
^ hj — I , hj ^0 , y = 1, 2,- • *, WZ2
where . r^, is probability of selecting R^,R2> . of
countermeasure R ( where R is acceleration mean square
root ) , and h,,h^, . X, is probability of selecting
. , of countermeasure H ( where H is
mean square root of deflection or handling and
satiability ) .
It is called hybrid game method while these probability is
leaded into the method . Countermeasure R selects in order
to get maximization of minimization paying expected value
of column vector of paying matrix , and countermeasure H
selects hj in order to get minimization of maximization
paying expected value of row vector of paying matrix .
If rank of paying matrix is x ,R should select r., as
follows.
max\min\
(5)
relative
And H should select hj as follows :
MAXl MINI
j=i
(6)
relative hj
a.j (z = l, . ,y = l, . ,^2) in (5) and (6) is element
value of paying matrix , basing vehicle theory it can be got
as follows :
a.j = C, ! + (7)
Where C^,C^ in (7) is coefficient of paying matrix ( The
paper orders they is 1 as an example , as Q ,C2 ’ s real value
about very much condition is related to some privacy
1198
problem it can’ t be introduced. ) . (5), (6) called respectively
minimization maximization expected value and maximization
minimization expected value can be abbreviated as
MAxi^IN^^ )} and MIN ^AX(^ )} . if r.,andhj is
got as optimization of countermeasure , it can be wntten as
follows :
MAX^IN^ ^^Optimization Countermeasure
Expected Value ^ MIN
A probability association {r^ Xj) can be content with
optimization expected value as follows :
Optimization Countermeasure Expected
m^ m-i
Value=|] J^aij»r;»hj (9)
/=!
Writer advances reformation simplex algorithm for resolving
the game problem as reference [1] . In the paper the water
selects only an example to introduce calculation results as
follows because the paper has limited space .
6 Results of Game Optimization Method of Control Law
Paying matrix as fig. 3 and fig. 4 , optimization result of
control law of a truck is got using above game method as
figure 5 and 6 , calculated truck’s main parameters as
follows .
Wheel distance is 1.23 meter , axle distance is 3.6 meter ,
mass is 13880.0 kg , front tyre and axle’ s mass is 280.0 kg ,
rear tyre and axle’ s mass is 280.0 kg , X axis’ rotational
inertia is 1935 kg^m^ , Y axis’ rotational inertia is 710
kg-m- .
In fig. 3 and 4 3(IJ) is paying matrix value
{i = 1, . ,m, J = 1, . ,m,) . In fig. 5 and fig. 6 , K1 of
RMSMIN and K3 of RMSMIN are respectively front and
rear suspension stiffness of getting optimization
countermeasure expected value of above game method , it is
changing while road surface rough coefficient Q and
automobile velocity V is changing . K1 and K3 also rise
1199
when velocity V rises . This is called control law of active
suspension . The result in fig. 5 and fig. 6 has been verified
by road test .
Fig. 3 paying matrix Fig. 4 paying matrix
7 Conclusion
The paper introduces two method to get optimization control
Fig. 5 front suspension Fig. 6 rear suspension
optimization stiffness optimization stiffness
law of active suspension , and the control law is verified to
ability to be applied to real control of active suspension . This
will develop control accuracy and speed of active
suspension .Off course , it need being researched further .
8 reference
1 Lei Yucheng , Theory and Engineering Realization of
Semi — Active Control of Vehicle Vibration ,Dr. paper ,
Harbin Institute of Technology , China ,1995 6 .
2 Thompson A.G. , A Suspension Proc. Int of Mechanical
1200
Engr. Vol 185 No. 36, 970 — 990,553 563 .
3 Lei , S. , Fasuda , E. and Hayashi , Y. : “An Experimental
Study of Optimal Vibration Adjustment Using Adaptive
Control Methods ”, Proc . IMechE Int . Conf . Advanced
Suspensions , London , England , (1988) , C433/88 , 119-
124.
4 Kamopp D. , Active Damping in Road Vehicle Systems ,
VSD, 12(1983), 291-316.
5 Kamopp D . C ., Grosby M. J. & Harword R. ,Vibration
Control Using Semi-Active Force , Generator , Trans .
ASME, J . Eng . for Ind . Vol . 96 Ser . B , No .2 , (1974) ,
619-626 .
1201
1202
Designing Heavy Truck Suspensions for Reduced Road Damage
Mehdi Ahmadian
Edward C. Mosch Jr.
Department of Mechanical Engineering
Virginia Polytechnic Institute and State University
Blacksburg, Virginia 24061-0238; USA
(540) 231-4920/-9100(fax)
ahmadian@vt.edu
ABSTRACT
The role of semiactive dampers in reducing tire dynamic loading is examined.
An alternative to the well-known skyhook control policy, called
“groundhook,” is introduced. Using the dynamic model of a single suspension,
it is shown that groundhook semiactive dampers can reduce tire dynamic
loading, and potentially lessen road damage, for heavy trucks.
INTRODUCTION
The main intent of this work is to determine, analytically, the role of
semiactive suspension systems in reducing tire dynamic loading, and road and
bridge damage. Although primary suspension systems with semiactive
dampers have been implemented in some vehicles for improving ride and
handling, their impact on other aspects of the vehicle remain relatively
unknown. Specifically, it is not yet known if implementing semiactive
dampers in heavy truck suspension systems can reduce the tire dynamic
forces that are transferred by the vehicle to the road. Reducing dynamic forces
will result in reducing pavement loading, and possibly road and bridge damage.
The idea of semiactive dampers has been in existence for more than
two decades. Introduced by Karnopp and Crosby in the early 70’ s [1-2],
semiactive dampers have most often been studied and used for vehicle primary
suspension systems. A semiactive damper draws small amounts of energy to
operate a valve to adjust the damping level and reduce the amount of energy
that is transmitted from the source of vibration energy (e.g., the axle) to the
suspended body (e.g., the vehicle structure). Therefore, the force generated by
a semiactive damper is directly proportional to the relative velocity across the
damper (just like a passive damper). Another class of dampers that is usually
considered for vibration control is fully active dampers. Active dampers draw
1203
relatively substantial amounts of energy to produce forces that are not
necessarily in direct relationship to the relative velocity across the damper.
The virtues of active and semiactive dampers versus passive dampers
have been addressed in many studies [3-10]. Using various analytical and
experimental methods, these studies have concluded that in nearly all cases
semiactive dampers reduce vibration transmission across the damper and
better control the suspended (or sprung) body, in comparison to passive
dampers. Further, they have shown that, for vehicle primary suspension
systems, semiactive dampers can lower the vibration transmission nearly as
much as fully active dampers; without the inherent cost and complications
associated with active dampers. This has led to the prototype application,
and production, of semiactive dampers for primary suspensions of a wide
variety of vehicles, ranging from motorcycles, to passenger cars, to bus and
trucks, and to military tanks, in favor of fully active systems.
Although there is abundant research on the utility of semiactive
dampers for improving vehicle ride and handling, their potential for reducing
dynamic forces transmitted to the pavement remains relatively unexplored.
This is because most suspension designers and researchers are mainly
concerned with the role of suspension systems from the vehicle design
perspective. Another perspective, however, is the effect of suspension
systems on transmitting dynamic loads to the pavement.
ROAD DAMAGE STUDIES
Dynamic tire forces, that are heavily influenced by the suspension, are
believed to be an important cause of road damage. Cole and Cebon [11]
studied the design of a passive suspension that causes minimum road damage
by reducing the tire force. They propose that there is a stronger correlation
between the forth power of the tire force and road damage than the dynamic
load coefficient (DLC) and road damage.
A simple measure of road damage, introduced by Cebon m [12], is the
aggregate fourth power force defined as
Na
Al=Y,Pjk k= 1,2,3 ..ns (1)
where Pjk = force applied by tire j to point k along the wheel path,
ria - number of axles on vehicle, and
Us = number of points along the road..
1204
DLC is a popular measurement frequently used to characterize dynamic
loading and is defined as the root mean square (RMS) of the tire force divided
by the mean tire force, which is typically the static weight of the vehicle. The
equation takes the form;
RMS Dynamic Tire Force
Static Tire Force
This study shows that minimum road damage, for a two degree-of-freedom
model, is achieved by a passive system with a stiffness of about one fifth of
current air suspensions and a damping of about twice that typically provided.
In practice, however, reducing the suspension stiffiiess can severely limit the
static load carrying capacity of the suspension and cause difficulties in vehicle
operation. Further, higher damping can substantially increase vibration
transmission to the body and worsen the ride.
In another study by Cole and Cebon [13] a two-dimensional articulated
vehicle simulation is validated with measurements from a test vehicle. The
effect of modifications to a trailer suspension on dynamic tire forces are
investigated. The RMS of dynamic loads generated by the trailer are predicted
to decrease by 3 1 per cent, resulting in a predicted decrease in theoretical road
damage of about 13 per cent.
Yi and Hedrick compared the effect of continuous semiactive and
active suspensions and their effect on road damage using the vehicle simulation
software VESYM [14]. A control strategy based on the tire forces in a heavy
truck model is used to show that active and semiactive control can potentially
reduce pavement loading. They, however, mention that measuring the tire
forces poses serious limitation in practice.
The primary purpose of this paper is to extend past studies on
semiactive suspension systems for reducing road damage. An alternative
semiactive control policy, called "groundhook,” is developed such that it can
be easily applied in practice, using existing hardware for semiactive
suspensions. A simulation model representing a single primary suspension is
used to illustrate the system effectiveness. The simulation results show that
groundhook control can reduce the dynamic load coefficient and fourth power
of tire force substantially, without any substantial increase in body
acceleration.
1205
MATHEMATICAL FORMULATION
We consider a model representing the dynamics of a single primary suspension
in a heavy truck, as shown in Figure 1.
Truck Mass
Suspension
Stiffness
Suspension
and Tire Mass
Tire
Stiffness
Suspension
Damping
Road Input
Figure 1. Primary Suspension Model
This model has been widely used in the past for automobile applications, due
to its effectiveness in analyzing various issues relating to suspensions.
Although it does not include the interaction between the axles and the truck
frame dynamics, the model still can serve as an effective first step in studying
fundamental issues relating to truck suspensions. Follow up modeling and
testing, using a full vehicle, is needed to make a more accurate assessment.
The dynamic equations for the model in Figure 1 are:
M^x^+C(x^-X2) + K(x^~X2) = 0 (3a)
MjX, - C(ii -X2)-K{x^ - ^2) + 0
The variables Xi and X2 represent the body and axle vertical displacement,
respectively. The variable Xjn indicates road input, that is assumed to be a
random input with a low-pass (0 - 25 Hz) filter. The amplitude for Xjn is
adjusted such that it creates vehicle and suspension dynamics that resembles
field measurements. Such a function has proven to sufficiently represent
actual road input to the vehicle tires.
Table 1 includes the model parameters, that are selected to represent a
typical laden truck used in the U.S. The suspension is assumed to have a
linear stiffness in its operating range. The damper characteristics are modeled
as a non-linear function, as shown in Figure 2.
1206
Table 1. Model Parameters
Description
Symbol
Value
Body Mass
Ml
287 kg
Axle Mass
M2
34 kg
Suspension Stiffness
K1
196,142 N/m
Tire Stifftiess
Kt
1,304,694 N/m
Suspension Damping
C
See Table 2
The bilinear function in Figure 2 represents the force-velocity
characteristics of an actual truck damper. The parameters selected for both
passive and semiactive dampers are shown in Table 2. These parameters are
selected based on truck dampers commonly used in the U.S. Although we
examined the effect of damper tuning on dynamic loading, if falls outside the
scope of this paper. Instead, we concentrate here on comparing different
semiactive dampers with a passive damper, using the baseline parameters
shown in Table 2. The semiactive and passive damper characteristics used for
this study are further shown in Figure 3.
1207
Force Velocity Curve
Figure 3. Passive and Semiactive Damper Characteristics
SKYHOOK CONTROL POLICY
As mentioned earlier, the development of semiactive dampers dates back to
early 70’s when Kamopp and Crosby introduced the skyhook control policy.
For the system shown in Figure 1, skyhook control implies
X,(X] - ;C2 ) - 0 C = Con
X](X^-X2)<0 C = Coff
Where x, and represents the velocities of Mi (vehicle body) and M2 (axle),
respectively. The parameters Con and Coff represent the on- and off-state of
the damper, respectively, as it is assumed that the damper has two damping
levels. In practice, this is achieved by equipping the hydraulic damper with an
orifice that can be driven by a solenoid. Closing the orifice increases damping
level and achieves Con, whereas opening it gives Coff .
1208
Table 2. Damper Parameters
Passive
Semiactive On-State
Semiactive Off-State
n
0.25
0.35
0.03
m
0.10
0.15
0.03
Vbb
0.254 m/sec
0.254 m/sec
0.254 m/sec
B
0.20
0.30
0.03
S4
0.10
0.15
0.03
Vbj
0.254 m/sec
0.254 m/sec
0.254 m/sec
The switching between the two damper states, shown in Eq. (4), is
arranged such that when the damper is opposing the motion of the sprung
mass (vehicle body), it is on the on-state. This will dampen the vehicle body
motion. When the damper is pushing into the body, it is switched to the off-
state to lower the amount of force it adds to the body. Therefore, a semiactive
damper combines the performance of a stiff damper at the resonance
frequency, and a soft damper at the higher frequencies, as shovm in Figure 4.
1209
Figure 5. Groundhook Damper Configurations: a) optimal groundhook
damper configuration, b) semiactive groundhook damper
configuration.
This feature allows for a better control of the vehicle body, as has been
discussed in numerous past studies. The skyhook control policy in Eq. (4),
however, works such that it increases axle displacement, X2, (commonly called
wheel hop). Because the tire dynamic loading can be defined as
DL = KtX2 (5)
The skyhook control actually increases dynamic loading. As mentioned earlier
the development of skyhook policy was for improving ride comfort of the
vehicle, without losing vehicle handling. Therefore, the dynamic loading of the
tires was not a factor in the control development.
GROUNDHOOK CONTROL POLICY
To apply semiactive dampers to reducing tire dynamic loading, we propose an
alternative control policy that can be implemented in practice using the same
hardware needed for the skyhook policy. To control the wheel hop, this
policy, called “groundhook,” implies:
X, (;ci - X2 ) < 0 C = Con
(6a)
x, (Xt - ^2 ) > 0 C = Coff
(6b)
As shovm in Figure 5, the above attempts to optimize the damping force on
the axle, similar to placing a damper between the axle and a fictitious ground
(thus, the name “groundhook”). The groundhook semiactive damper
maximizes the damping level (i.e., C = Con) when the damper force is opposing
1210
the motion of the axle; otherwise, it minimizes the damping level (i.e., C =
Coff). The damper hardware needed to implement groimdhook semiactive is
exactly the same as
the skyhook semiactive, except for the control policy programmed into the
controller.
SIMULATION RESULTS
The model shown in Figure 1 is used to evaluate the benefits of groundhook
dampers versus passive and skyhook dampers. A non-linear damper model
was considered for the simulations, as discussed earlier. The road input was
adjusted such that the dynamic parameters for the passive damper resembles
actual field measurements. Five different measures were selected for
comparing the dampers:
• Dynamic Load Coefficient (DLC)
• Fourth Power of the tire dynamic load
• Sprung mass acceleration
• Rattle Space (relative displacement across the suspension)
• Axle Displacement, relative to the road
Dynamic load coefficient and fourth power of tire force are measures
of pavement dynamic loading and are commonly used for assessing road
damage. They are both considered here because there is no clear consensus on
which one is a better estimate of road damage. Axle displacement, relative to
the road, indicates wheel hop and is directly related to DLC and tire force,
therefore it is yet another measure of road damage. Sprung mass acceleration
is a measure of ride comfort. Our experience, however, has shown that for
trucks this may not be a reliable measure of the vibrations the driver feels in
the truck. The relative displacement across the dampers relates to the rattle
space, that is an important design parameter in suspension systems,
particularly for cars. For each of the above measures, the data was evaluated
in both time (Figures 6-7) and frequency domain (Figures 8 - 10). In time
domain, the root mean square (RMS) and maximum of the data for a five-
second simulation are compared. In frequency domain, the transfer function
between each of the measures and road displacement is plotted vs. frequency.
The frequency plots highlight the effect of each damper on the body and axle
resonance frequency.
Figures 6 and 7 show bar charts of root mean square (RMS) and
maximum time data, respectively. In each case the data is normalized with
respect to the performance of passive dampers commonly used in trucks.
Therefore, values below line 1 .0 can be interpreted as an improvement over the
existing dampers. As Figures 6 and 7 show, groundhook dampers significantly
1211
improve pavement loading, particularly as related to the fourth power of tire
force. Furthermore, the rattle space is improved slightly over passive
dampers, indicating that groundhook dampers do not impose any additional
burden on the suspension designers.
One measure that has increased due to groundhook dampers is body
acceleration. As mentioned earlier, in automobiles this measure is used as an
indicator of ride comfort. In our past testing, however, we have found that for
trucks it is a far less accurate measure of ride comfort. This is mainly due to
the complex dynamics of the truck frame and the truck secondary suspension.
A more accurate measure of ride comfort is acceleration at the B-Post (the
post
Tire Axle Body Rattle Space
Dynamic Displacment Acceleration
Force ^ 4
Figure 6. RMS Time Data Normalized with respect to Passive Damper
Tire Axle Body Rattle Space
Dynamic Displacment Acceleration
Force 4
Figure 7. Max. Time Data Normalized with respect to Passive Damper
1212
behind the driver), which cannot be evaluated from the single suspension
model considered here. Nonetheless, the body acceleration is included for the
sake of completeness of data.
The model shows that skyhook dampers actually increase the
measures associated with pavement loading, while improving body
acceleration. This agrees with the purpose of skyhook dampers that are
designed solely for improving the compromise between ride comfort and
vehicle handling. The improvement in ride comfort occurs at the expense of
increased pavement loading.
Figure 8. Transfer Function between Axle Displacement and Road Input
Figures 8-10 show the frequency response of the system due to each damper.
In each figure, the transfer fimction between one of the measures and input
displacement is plotted vs. frequency. These plots highlight the impact of
skyhook and groundhook on the body and wheel hop resonance, relative to
existing passive dampers. The frequency plots indicate that the
1213
Transfer
Frequency (Hz)
groundhook dampers reduce axle displacement and fourth power of tire
dynamic force at wheel hop frequency. At body resonance frequency,
groundhook dampers do not offer any benefits over passive dampers. The
frequency results for body acceleration and rattle space are similar to those
discussed earlier for the time domain results. The frequency plots show that
the skyhook dampers offer benefits over passive dampers at frequencies close
to the body resonance frequencies. At the higher frequencies, associated with
wheel hop, skyhook dampers result in a larger peak than either passive or
groundhook dampers. This indicates that skyhook dampers are not suitable
for reducing tire dynamic loading.
CONCLUSIONS
An alternative to skyhook control policy for semiactive dampers was
developed. This policy, called “groundhook,” significantly improves both
dynamic load coefficient (DLC), and fourth power of tire dynamic load,
therefore holding a great promise for reducing road damage to heavy trucks.
The dynamic model used for assessing the benefits of groundhook dampers
represented a single suspension system. Although the results presented here
show groundhook dampers can be effective in reducing tire dynamic loading
and pavement damage, more complete models and road testing are necessary
for more accurately assessing the benefits.
REFERENCES
1. Crosby, M. J., and Karnopp, D. C., "The Active Damper," The Shock
and Vibration Bulletin 43, Naval Research Laboratory, Washington, D.C.,
1973.
2. Karnopp, D. C., and Crosby, M. J., "System for Controlling the
Transmission of Energy Between Spaced Members," U.S. Patent
3,807,678, April 1974.
3. Ahmadian, M. and Marjoram, R. H., “Effects of Passive and Semi-active
Suspensions on Body and Wheelhop Control,” Journal of Commercial
Vehicles, Vol. 98, 1989, pp. 596-604.
4. Ahmadian, M. and Marjoram, R. H., “On the Development of a
Simulation Model for Tractor Semitrailer Systems with Semiactive
Suspensions,” Proceedings of the Special Joint Symposium on Advanced
Technologies, 1989 ASME Winter Annual Meeting, San Francisco,
California, December 1989 (DSC-Vol. 13).
5. Hedrick, J. K., "Some Optimal Control Techniques Applicable to
Suspension System Design," American Society of Mechanical Engineers,
Publication No. 73-ICT-55, 1973.
1215
6. Hac, A., "Suspension Optimization of a 2-DOF Vehicle Model Using
Stochastic Optimal Control Technique," Journal of Sound and Vibration,
1985.
7. Thompson, A. G., "Optimal and Suboptimal Linear Active Suspensions
for Road Vehicles," Vehicle System Dynamics, Vol. 13, 1984.
8. Kamopp, D., Crosby, M. J., and Harwood, R. A., "Vibration Control
Using Semiactive Force Generators," American Society of Mechanical
Engineers, Journal of Engineering for Industry, May 1974, pp. 619-626.
9. Krasnicki, E. J., "Comparison of Analytical and Experimental Results for
a Semiactive Vibration Isolator," Shock and Vibration Bulletin, Vol. 50,
September 1980.
10. Chalasani, R.M., "Ride Performance Potential of Active Suspension
Systems-Part 1: Simplified Analysis Based on a Quarter-Car Model,"
proceedings of 1986 ASME Winter Annual Meeting, Los Angeles, CA,
December 1986.
11. Cole, D. J. and Cebon, D., “Truck Suspension Design to Minimize Road
Damage,” Proceedings of the Institution of Mechanical Engineers, Vol.
210, D06894, 1996, pp. 95-107.
12. Cebon, D., “Assessment of the Dynamic Forces Generated by Heavy
Road Vehicles,” ARRB/FORS Symposium on Heavy Vehicle Suspension
Characteristics, Canberra, Australia, 1987.
13. Cole, D. J. and Cebon, D., “Modification of a Heavy Vehicle Suspension
to Reduce Road Damage,” Proceedings of the Institution of Mechanical
Engineers, Vol. 209, D03594, 1995.
14. Yi, K. and Hedrick, J. K., “Active and Semi- Active Heavy Truck
Suspensions to Reduce Pavement Damage,” SAE SP-802, paper 892486,
1989.
1216
Active Vibration Control of Isotropic Plates Using
Piezoelectric Actuators
A. M. Sadri", J. R., Wright* and A. S. Cherry*
The Manchester School of Engineering, Manchester M13 9PL, UK
and
R. J. Wynnes
Sheffield Hallam University, School of Engineering, Sheffield, UK
Abstract: Theoretical modelling of the vibration of plate components of a
space structure excited by piezoelectric actuators is presented. The equations
governing the dynamics of the plate, relating the strains in the piezoelectric
elements to the strain induced in the system, are derived for isotropic plates
using the Rayleigh-Ritz method. The developed model was used for a simply
supported plate. The results show that the model can predict natural
frequencies and mode shapes of the plate very accurately. The open loop
frequency response of the plate when excited by the patch of piezoelectric
material was also obtained. This model was used to predict the closed loop
frequency response of the plate for active vibration control studies with
suitable location of sensor-actuators.
Introduction
Vibration suppression of space structures is very important because
they are lightly damped due to the material used and the absence of air
damping. Thus the modes of the structure must be known very
accurately in order to be affected by the controller while avoiding
spillover. This problem increases the difficulty of predicting the
behaviour of the structure and consequently it might cause unexpected
on-orbit behaviour.
These difficulties have motivated researchers to use the
actuation strain concept. One of the mechanisms included in the
actuation strain concept is the piezoelectric effect whereby the strain
induced through a piezoelectric actuator is used to control the
Research Student, Dynamics &: Control Research Group.
^ Professor , Dynamics & Control Research Group,
^ Former Lecturer, Dynamics & Control Research Group.
^ Professor of Mechanical and Control Engineering.
1217
deformation of the structure [1]. It can be envisaged that using this
concept in conjimction with control algorithms can enhance the ability
to suppress modes of vibration of flexible structures.
Theoretical and experimental results of the control of a flexible
ribbed antenna using piezoelectric materials has been investigated in
[2]. An active vibration damper for a cantilever beam using a
piezoelectric polymer has been designed in [3]. In this study,
Lyapunov's second or direct method for distributed-parameter systems
was used to design control algorithms and the ability of the algorithms
was verified experimentally. These works have clearly shown the
ability of piezoelectric actuators for vibration suppression. However,
they have been limited to one dimensional systems. Obviously, there is
a need to understand the behaviour of piezoelectric actuators in two
dimensional systems such as plates.
Vibration excitation of a thin plate by patches of piezoelectric
material has been investigated in [4]. Their work was basically an
extension of the one dimensional theory derived in [1] to show the
potential of piezoelectric actuators in two dimensions. In their studies,
it was assumed that the piezoelectric actuator doesn't significantly
change the inertia, mass or effective stiffness of the plate. This
assumption is not guaranteed due to the size, weight and stiffness of
the actuator. Based on this assumption, their model can not predict the
natural frequencies of the plate accurately after bonding piezoelectric
actuators. Therefore, it is essential to have a more general model of a
plate and bonded piezoelectric actuators with various boundary
conditions. The model should be able to predict frequency responses
because this is fundamental to the understanding of the behaviour of
the system for control design purposes. It is the objective of the current
study to develop such a modelling capability.
Previous work [5, 6, 7] has concentrated on the modelling and
control of a cantilever beam. The method used involved bonding
piezoelectric material to a stiff constraining layer, which was bonded to
the beam by a thin viscoelastic layer in order to obtain both active and
passive damping. Then a Rayleigh-Ritz model was developed and
used to derive a linearized control model so as to study different
control strategies. In the work described in this paper, the method has
been extended to the more complex plate problem. The paper
introduces a modelling approach based on the Rayleigh-Ritz assumed
mode shape method to predict the behaviour of a thin plate excited by
a patch of piezoelectric material bonded to the surface of the plate. The
model includes the added inertia and stiffness of the actuator and has
been used to predict the frequency response of the plate. Suggestions
for future work are also included.
1218
Theoretical Modelling
In developing the Rayleigh-Ritz model of a plate excited by a
patch of piezoelectric material bonded to the surface of the plate, a
number of assumptions must be made. The patch of piezoelectric
material is assumed to be perfectly bonded to the surface of the plate.
The magnitude of the strains induced by the piezoelectric element is a
linear fimction of the applied voltage that can be expressed by
e:=E:
(1)
Here is the piezoelectric strain constant, is the piezoelectric layer
thickness and V33 is the applied voltage. The index 31 shows that the
induced strain in the '1' direction is perpendicular to the direction of
poling '3' and hence the applied field. The piezoelectric element
thickness is assumed to be small compared to the plate thickness. The
displacements of the plate middle surface are assumed to be normal to
it due to the bending affects.
Figure 1 shows the configuration of the bonded piezoelectric
material relative to the surface of the plate.
w
Figure 1. Configuration of the bonded piezoelectric actuator on the surface of the
plate.
In figure 1, 4 and 4 are the dimensions of the plate, x, , X2 , y, and y^
are the boundaries of the piezoelectric element and w, v and w are the
displacements in the x , y and z direction, respectively.
To derive the equations of motion of the plate based on the
Rayleigh-Ritz method, both the strain energy U and kinetic energy T of
the plate and the piezoelectric element must be determined. The strain
1219
and kinetic energy result from the deformation produced by the
applied strain which is induced by exciting the piezoelectric element.
The deformations can be expressed by the combination of the midplane
displacement and the deformations resulting from the bending of the
plate.
Strain Energy
The strain energy of the plate and piezoelectric material can be
calculated by
U = +T;«yY,<y)dVp +T^T„)dVp. (2)
where 8 is the inplane direct strain, a is the inplane direct stress, t is
the inplane shear stress and y is shear strain. dV shows volume
differential and indices p and pe refer to the plate and piezoelectric
actuator, respectively. The strains 8^ , 8^ and y can be shown to be
For the Plate :
du
dv d (3)
_du dv d^w
^ dy dx dxdy
For the Piezoelectric actuator :
Ev =
du
a
^~'dx~
^ a?*
Sv
a
du av
« ■ _ 1 ^
dy dx
'"33=S,-H'^33
a
dxdy ^
(4)
where _ refers to the strains due to the deformation. The stresses ,
Gy , y can be expressed as
1220
For the Plate :
E„
1
0
‘e. '
1
0
y.vy.
0
0
r.y_
2 .
(5)
For the Piezoelectric actuator :
'
0^.
II
1
0
Sx-H^33
1
0
r.y_
0
0
l-v
2 .
. Is
(6)
where E is Young modulus and v is Poisson's ratio for the assumed
isotropic material. Substituting equations 3, 4, 5 and 6 into 2 yields the
strain energy of the plate and piezoelectric actuator.
^ S, + sj + 1 ( 1 - Dp) dVp
- '"''33 )' + 2Dpp( e, - HV33 ){ E, - M.V33 )^ + ^ 1 ciVpp.
•'Vp, ^ _ u — - tC
(7)
Kinetic Energy
To obtain the kinetic energy, the velocity components in x, y and
z directions are needed. The velocity components can be calculated by
differentiating the displacement components which are
dw
u— u— z—
dx
dw
(8)
dy
w=w.
Differentiating equations 8 yields
1221
dw
u= u- z —
ax
dw
v= V- z —
dy
w
(9)
where u , y and w are the velocity components in the x, y and z
directions respectively. Using these velocity components, the kinetic
energies of the plate and piezoelectric actuator are obtained as
dw
P rw^ + (u-z^f + (v-z^^]dV^
9x
ay'
(10)
where p is the mass density.
Equation of Motion
The static or dynamic response of the plate excited by the
piezoelectric actuator can be calculated by substituting the strain and
kinetic energy into Lagrange's equation
d dT dT ^ dU
dt dq. dq,"^ dq.
(11)
where q^ represents the ith generalised coordinate and is the ith
generalised force. As there are no external forces (the force applied by
5\e piezoelectric element is included as an applied strain) or gyroscopic
terms and there is no added damping, Lagrange's equation reduces to :
dt dq, ^ dq,
(12)
Now the equation of motion can be obtained by using the expression
obtained for the strain and kinetic energy, and the assumed shape
functions for flexural and longitudinal motion
1222
u(x,y,t)~
v(x,y,t)={i,(x,y)y
w(x, y, t) = {(|)(j:, y)}^ {?(?)}.
(13)
Here y , ^ and ([) are the assumed displacement shape and h , f and g
are generalised coordinate of the plate response m x, y and z
directions. Using the shape functions expressed in equations 13,
substituting equations 7 and 10 into equation 12, and including
Rayleigh damping yields the equation of motion of the plate in the
form
where M, C and K are mass, damping and stiffness matrices and P is
the voltage-to-force transformation vector. Vector q represents the
plate response modal amplitudes and V is the applied voltage.
State-Space Equations
A model of a structure found via finite element or Rayliegh-Ritz
methods results in second-order differential equations of the form
[M]{9}+[C]{?}+[if]{?}={P}V (15)
Choosing state variables x^_ = q and = i, , equation 1 may be
reduced to a state-space representation as follows :
q = x =X2
/ / / (16)
q = X2= ~M-^Kq - M~'Cq + M~'PV.
Equations 15 can then be rewritten as
r 0 /I
0
1
1
1
1
A.
+
^2.
.^2_
[,] = [/ 0]
(17)
where [ ], { } are ignored. It should be noted that the vector q must be
multiplied by the shape fimctions to produce the actual displacement.
1223
Results
The model was used to investigate the response of a simply
supported plate. In order to maintain symmetry of the geometric
structure a piezoelectric actuator is assumed to be bonded to both the
top and bottom surfaces of the plate. So The symmetry of the elements
causes no extension of the plate midplane and the plate deforms in
pure bending. In this case the shape functions are assumed to be :
\{/(x,y) = 0
^(x,y) = 0 (18)
41 ( X, y ) = jinf x)sin( ^ y)
A-
where m and n are the number of half waves in the x and y directions.
The properties of the plate are given in Table 1 and its
dimensions are 4 = Q38m , 4, = Q30/w and 4=i.5876m/K. Tables 2 and 3
show the natural frequencies of the bare plate obtained by the thin
plate theory and the RR model, respectively. Since the shape functions
used in this example express the exact shape of the simply supported
plate, the natural frequencies included in Tables 2 and 3 are very close.
In order to excite the plate, a piezoelectric actuator with
configuration x, = 0.32 Xj = 0.36 m, = 0.04 OT and = 026 m is used above
and below. The natural frequencies of the plate after bonding the
piezoelectric actuator to the surface are given in Table 4. The results
show an increase in natural frequencies, showing that the added
stifness is more important than the added inertia.
Table 1 : Properties of the plate
207 7870 .292
Table 2 : Plate natural frequencies (rad / s) , Thin Plate Theory
1
2
3
4
1
437.5
1246.0
2593.5
4480.0
2
941.4
1749.9
3097.4
4983.9
3
1781.2
2589.7
3937.2
5823.7
4
2957.0
3765.5
5113.0
6999.5
1224
Table 3 : Plate natural frequencies (rad / s) , RR Model
/n
1
J
2
3
4
1
437.5
1245.9
2593.2
4479.1
2
941.3
1749.7
3097.0
4982.8
3
1781.1
2589.4
3936.6
5822.3
4
2956.7
3764.9
5111.9
6997.4
Table 4 : Plate-Piezo natural frequencies (rad / s)
/n
1
2
3
4
1
444.0
1257.1
2611.3
4502.8
2
957.2
1775.6
3182.1
5076.7
3
1854.6
2642.5
4072.3
6029.6
4
3076.6
3933.8
5224.6
7277.2
Initially, the piezoelectric actuator was excited by a constant DC
voltage. The result of this action is shown in figure 2 which shows the
dominant out of plane displacement around the location of the
piezoelectric actuator bonded to the surface of the plate. To show the
modes of vibration, the piezoelectric actuator was excited by a voltage
with frequencies near to the natural frequencies of modes (2,2) and
(1,3). The response in figures 3 , 4 show that the piezoelectric actuator
excited both of these modes.
Displacement Distribution (X-Y)
0 0
Figure 2. Static Displacement
1225
Displacement Distribution (X-Y)
0 0
Figure 3. Vibration of the plate, mode (2,2)
Displacement Distribution (X-Y)
0 0
Figure 4. Vibration of the plate, mode (3,1)
The frequency response of tihe plate at the centre was obtained
by exciting the piezoelectric actuator at a range of frequencies between
0 and 4000 rad/s. Figure 5 shows the frequency response of the plate
at its centre. The frequency response of the plate at y = 0.5L^ along the
x-direction is shown in figure 6. It can be seen that the amplitude of
vibration of some modes are very high compared to that of the other
modes. Special attention must be given for the suppression of vibration
of these modes.
1226
Frequency Response of the Plate
Figure 5. Frequency response of the plate at the centre
Frequency Response of the Plate
4000
Figure 6. Frequency response of the plate along x-direction
The results show that it is possible to predict the frequency
response of a plate when it is excited by a patch of piezoelectric
material. Consequently, a sensor model can be also added to the model
and a signal proportional to velocity fed back to the piezoelectric
patch. As a result, the closed loop frequency response of the plate can
be obtained theoretically which is very important for active vibration
control studies. This also permits the investigation of the optimal
1227
location of the actuators and the study of control algorithms for the
best possible vibration suppression before using any costly
experimental equipment.
For this purpose, two patches of piezoelectric (lOcmxScm and
5cmx4cm), whose specifications are listed in table 5, were bonded to
the surface of the plate in different locations and then the plate was
excited by a point force marked by "D" in the figure 7. In figure 7 the
dash lines are showing the nodal lines of a simply supported plate up
to mode (3, 3).
Table 5 : Properties of the actuator
(mm) EJxlO''^N/m')pJkg/m') dJxlO'^'mfv) ^
.2 6.25 7700 -180 .3
An actuator is most effective for control of a particular mode if
the sign of the strain due to the modal deflection shape is the same
over die entire actuator. Consequently, as can be seen from figure 7,
the actuators are placed between the nodal lines and at the points of
maximum curvature in order to obtain good damping effect on the
modes of interest. Then two accelerometers were located at the center
of the location of the actuators, marked by "S" in figure 7, in order to
have collocated sensor-actuators. The signals obtained by the
accelerometers are integrated and fed back to the actuators separately.
Therefore rate feedback was used in this configuration. This leads to
the feedback control law
V = kq
(19)
where k is an amplification factor or feedback gain. Substituting
equation 19 into equation 17 the closed-loop state-space representation
of the system can then be obtained as
0
-M-‘K
_ J
I —
1
M-^(C-kP)_
[^]=[/ 0]
(20)
1228
Then the effects of the actuators on vibration suppression were
investigated. At first, only the actuator near to the center of the plate
was used to suppress the vibration. The effect of this is shown in figure
8. As can be seen, damping in some modes are improved and some
modes are untouched.
The second actuator was added to the model to see its effect on
modes of vibration.
Figure 7. Plate with Bonded Piezoelectric Actuators
Frequency Response of the Plate
Figure 8. Open and Closed loop Frequency Response of the plate
solid line ; open loop, dash line : closed loop
1229
Frequency Response of the Plate
-50
_2qqI - 1 - 1 - 1 - 1 - 1 - 1 — — - 1 - 1
0 500 1000 1500 2000 2500 3000 3500 4000
Frequency (rad/s)
Figure 9. Open and Closed Loop Frequency Response of The Plate
solid line : open loop, dash line : closed loop
The open and closed loop frequency response of the plate when excited
by the point force and controlled by two actuators is shown in figure 9.
As can be seen, significant vibration suppression was obtained in both
lower and higher modes. Also, it shows that the place of actuators was
successfully chosen. This analysis showed that obtaining reasonable
but not necessarily optimal placement of actuators in structures is very
important in order to obtain a high level of damping in the modes of
interest. Obviously, bonding more than one piezoelectric actuator in
suitable locations helps to successfully suppress vibration of the plate.
Conclusions
A model of an active structure is fundamental to the design of
control strategies. It can be used to analyse the system and investigate
optimal control strategies without using costly experimental
equipment.
A Rayleigh-Ritz model has been developed to analyse the behaviour
of a thin plate excited by a patch of piezoelectric material. The model
has been used for a simply supported plate. It has been shown that the
model can predict natural frequencies of the plate alone very
accurately. The obtained mode shapes also correspond to the actual
mode shapes. The frequency response of the plate can be obtained to
show the suitability of the model for control design studies. This study
allowed the behaviour of the system in open and closed loop form for
active vibration control purposes to be investigated. Two piezoelectric
actuators were used to investigate their effectiveness on vibration
1230
suppression of the plate. The analysis showed that the location of two
actuators was very important to increase the level of damping in both
lower and higher frequency modes. The future work will be to extend
the model to analyse a plate with more than two patches of
piezoelectric material with optimal configuration, obtained by
controllability theory, and independent controller for vibration
suppression, and experimental verification of the theoretical analysis.
References
1. Crawley, E. F. and de Luis, J., Use of Piezoelectric Actuators as
Elements of Intelligent Structures, AlAA Journal, Vol. 25, No. 10,
1987, p. 1373.
2. Dosch, J., Leo, D. and Inman, D., Modelling and control for
Vibration Suppression of a Flexible Active structure, AIAA Journal
of Guidance, Control and Dynamics, Vol. 18, No. 2, 1995, p.340.
3. Bailey, T. and Hubbard J. E. Jr., Distributed Piezoelectric Polymer
Active Vibration Control of a Cantilever Beam, AIAA Journal of
Guidance, Control and Dynamics, Vol. 8, No. 4, 1985, p.605.
4. Dimitriadis, E. K., Fuller, C. R., Rogers C. A., Piezoelectric
Actuators for Distributed Vibration Excitation of Thin Plates,
Journal of Vibration and Acoustics, Vol. 113, No. 1, 1991, p. 100.
5. Azvine, B., Tomlinson, G. R. and Wynne, R. J., Use of Active
Constrained Layer Damping for Controlling Resonant, Journal of
Smart Materials and Structures, No. 4, 1995.
6. Rongong, J. A., Wright, J. R., Wynne, R. J. and Tomlinson, G. R.,
Modelling of a Hybrid Constrained Layer/Piezoceramic Approach
to Active Damping, Journal of Vibration and Acoustics, To appear.
7. Sadri, A. M., Wynne, R. J. and Cherry, A. S., Modelling and
Control of Active Damping for Vibration Suppression, UKACC
International Conference on Control' 96, 2-5 September 1996.
8. Bathe, K,, Finite Element Procedures in Engineering Analysis, Prentice-
Hall, Inc., 1982.
9. Blevins, R. D., Formulas for Natural Frequency and Mode Shapes, Van
Nostrand Remhold, NY 1979.
10. Thomson, W. T., Theory of Vibration with Applications, Prentice-Hall,
Inc., 1988.
1231
Active control of sound transmission into a rectangular
enclosure using both structural and acoustic actuators
S.M. Kim and MJ. Brennan
ISVR, University of Southampton, Highfieid, Southampton, S017 IBJ, UK
ABSTRACT
This paper presents an analytical investigation into the active control of sound
transmission in a ‘weakly coupled’ structural-acoustic system. The system
under consideration is a rectangular enclosure having one flexible plate
through which external noise is transmitted. Three active control systems
classified by the type of actuators are discussed. They are; i) a single force
actuator, ii) a single acoustic piston source, and iii) simultaneous use of both
the force actuator and the acoustic piston source. For all three control systems
the acoustic potential energy inside the enclosure is adopted as the cost
function to minimise, and perfect knowledge of the acoustic field is assumed.
The results obtained demonstrate that a single point force actuator is effective
in controlling well separated plate-controlled modes, whereas, a single
acoustic piston source is effective in controlling well separated cavity-
controlled modes provided the discrete actuators are properly located. Using
the hybrid approach with both structural and acoustic actuators, improved
control effects on the plate vibration together with a further reduction in
transmitted noise and reduced control effort can be achieved. Because the
acoustic behaviour is governed by both plate and cavity-controlled modes in a
‘weakly coupled’ structural-acoustic system, the hybrid approach is desirable
in this system.
1. INTRODUCTION
Analytical studies of vibro-acoustic systems have been conducted by many
investigators to achieve physical insight so that effective active control
systems can be designed. It is well established that a single point force actuator
and a single acoustic piston source can be used to control well separated
vibration modes in structures and well separated acoustic modes in cavities,
respectively, provided that the actuators are positioned to excite these
modes [1,2]. Active control is also applied to structural-acoustic coupled
systems for example, the control of sound radiation from a piate[3-6] and the
sound transmission into a rectangular enclosure[7-8]. Meirovitch and
1233
Thangjitham[6], who discussed the active control of sound radiation from a
plate, concluded that more control actuators resulted in better control effects.
Pan et al[Z] used a point force actuator to control sound transmission into an
enclosure, and discussed the control mechanism in terms of plate and cavity-
controlled modes.
This paper is concerned with the active control of sound transmission into a
‘weakly coupled’ structural-acoustic system using both structural and acoustic
actuators. After a general formulation of active control theory for structural-
acoustic coupled systems, it is applied to a rectangular enclosure having one
flexible plate through which external noise is transmitted. Three active control
systems classified by the type of actuators are compared using computer
simulations. They are; i) a single force actuator, ii) a single acoustic piston
source, and iii) simultaneous use of both the force actuator and the acoustic
piston source. For all three control systems the acoustic potential energy inside
the enclosure is adopted as the cost function to minimise, and perfect
knowledge of the acoustic field is assumed. The effects of each system are
discussed and compared, and a desirable control system is suggested.
2. THEORY
2.1 Assumptions and co-ordinate systems
Consider an arbitrary shaped enclosure surrounded by a flexible structure and a
acoustically rigid wall as shown in Figure 1. A plane wave is assumed to be
incident on the flexible structure, and wave interference outside the enclosure
between the incident and radiated waves by structural vibration is neglected.
Three separate sets of co-ordinates systems are used; Co-ordinate x is used for
the acoustic field in the cavity, co-ordinate y is used for the vibration of the
structure, and co-ordinate r is used for the sound field outside the enclosure.
The cavity acoustic field and the flexible structure are governed by the linear
Helmholtz equation and the isotropic thin plate theory[9], respectively. The
sign of the force distribution function and normal vibration velocity are set to
be positive when they direct inward to the cavity so that the structural
contribution to acoustic pressure has the same sign as the acoustic source
contribution to acoustic pressure.
Weak coupling rather than full coupling is assumed between the structural
vibration system and the cavity acoustic system. Thus, the acoustic reaction
force on the strucural vibration under structural excitation and the structural
induced source effect on the cavity acoustic field under acoustic excitation is
neglected. This assumption is generally accepted when the enclosure consists
of a heavy structure and a big volume cavity. It is also assumed that the
coupled response of the system can be described by finites summations of the
1234
uncoupled acoustic and structural modes. The uncoupled modes are the rigid-
walled acoustic modes of the cavity and the in vacuo structural modes of the
structure. The acoustic pressure and structural vibration velocity normal to the
vibrating surface are chosen to represent the responses of the coupled system.
2.2 Structural-acoustic coupled response
The acoustic potential energy in the cavity is adopted as the cost function for
the global sound control, which is given by [2]
where, and Co respectively denote the density and the speed of sound in air,
and /?(x,ft)) is the sound pressure inside the enclosure.
The vibrational kinetic energy of the flexible structure, which will be used to
judge the control effect on structural vibration, is given by[l]
where, p., is the density of the plate material, h is the thickness of the plate.
If the acoustic pressure and the structural vibration are assumed to be
described by a summation of N and M modes, respectively, then the acoustic
pressure at position x inside the enclosure and the structural vibration velocity
at position y are given by
N
P(X,(B) =
n=I
M
ni=l
where, the N length column vectors ^ and a consist of the array of uncoupled
acoustic mode shape functions and the complex amplitude of the
acoustic pressure modes a^^(co) respectively. Likewise the M length column
vectors O and b consist of the array of uncoupled vibration mode shape
functions 0n,(y) and the complex amplitude of the vibration velocity modes
respectively.
The mode shape functions \f/^{x) and (l)Jy) satisfy the orthogonal property
in each uncoupled system, and can be normalised as follows.
V = lwlMdV (5)
S,=lfyy)dS (6)
1235
where, V and S/ are the volume of the enclosure and the area of the flexible
structure, respectively. Since mode shape functions are normalised as given by
(Eq. 5), the acoustic potential energy can be written as
£ =_L^a“a (7)
' 4p„c^
Similarly from (Eq. 2) and (Eq. 6), the vibrational kinetic energy can be
written as
£ =££^b«b (8)
2
Where superscript H denotes the Hermitian transpose.
For the global control of sound transmission, it is required to have knowledge
of the complex amplitude of acoustic pressure vector a for various excitations.
The complex amplitude of the n-th acoustic mode under structural and
acoustic excitation is given by[9, 10]
^1. (^) = 4 (^)(^ (x)‘y(x, co)dv + i//'„ (yMy, j (9)
where, s(x,Ci)) denotes the acoustic source strength density function in the
cavity volume V”, and w(y,6)) denotes the normal velocity of the surrounding
flexible structure on surface 5/-. The two integrals inside the brackets represent
the nth acoustic modal source strength contributed from s(x,Q)) and u(y,co) ,
respectively. The acoustic mode resonance term An(fi)) is given by
A,.(co) = - - - (10)
o}--Q)-+j2C„<a„(o
where a)„and C„ are the natural frequency and damping ratio of the nth
acoustic mode, respectively.
Substituting (Eq. 4) into (Eq. 9) and introducing the modal source strength
q,: = j ¥n X. 0))dV , then we get
2 / M
a„ (®) = A, (®) (®) + S O',,™ • b,„((0)
y V Hi=i
where, C„,,„ represents the geometric coupling relationship between the
uncoupled structural and acoustic mode shape functions on the surface of the
vibrating structure Sf and is given by[l 1]
c,,,„ = lvJy)<t>Jy’0>)ds (12)
If we use L independent acoustic control sources, can be written as
/=! \l /=!
1236
where = - j W ^ control source strength q,^i{co)
having an area of Scjj is defined at Xc,i.
Thus, the complex amplitude of acoustic modal pressure vector a can be
expressed as
a = Z.(D,q,+Cb) (14)
2
whereZa= -^A.
The matrix A is a (NxN) diagonal matrix in which each (n,n) diagonal term
consists of A„, the (NxM) matrix C is the structural-acoustic mode shape
coupling matrix, the (NxL) matrix Dq determines coupling between the L
acoustic source locations and the N acoustic modes, the L length vector (jc is
the complex strength vector of acoustic control sources, and b is the complex
vibrational modal amplitude vector. The {NxN) diagonal matrix Za can be
defined as the uncoupled acoustic modal impedance matrix which determines
the relationship between the acoustic source excitation and the resultant
acoustic pressure in modal co-ordinates of the uncoupled acoustic system.
Generally the impedance matrix is symmetric but non-diagonal in physical co¬
ordinates, however the uncoupled modal impedance matrix is diagonal
because of the orthogonal property of uncoupled modes in modal co-ordinates.
Since the flexible structure in Figure 1 is assumed to be governed by the
isotropic thin plate theory, the complex vibrational velocity amplitude of the
mth mode can be expressed as[10]
= + (15)
p,hSf ^
where, again p.v is the density of the plate material, h is the thickness of the
plate, is the area of flexible structure. Inside the integral /(y,ty) , p""\'^.(0) ,
and p‘"'(y,ty) denote the force distribution function, and the exterior and
interior sides of acoustic pressure distribution on the surface 5/ , respectively.
Because of the sign convention used, there is a minus sign in front of
(y, (O) . The structural mode resonance term Bm((0) can be expressed as
-(0^+j2^„^0),„co
where co^ and Cm natural frequency and the damping ratio of m-th
mode, respectively. Substituting (Eq. 3) into (Eq. 15), then we get
b,niO})=——B,„ico)\ g,,,„ico)+ g^,„i(o)-'£cl„-a,(co) (17)
p,hSf \ „=i ;
1237
where, L = L
• JSj- •'■i/
and Cl^ = C,„.. ■
If we use K independent point force actuators, the m-th mode generalised force
due to control forces, gc.m, can be written by
= XI <P,n(y)S(y-ycjc)dsf,j,ico) = X^/.».t/<.-.t(®)
k=\ ^ k=]
(18)
where 0,„(y)5(y-y.,t)rfS, and the k-th control point force l,{co)
JSj
is located at y^- it.
Thus the modal vibrational amplitude vector b can be expressed as
b = Y,(g,+Dff,-C’'a) (19)
where Ys = — ^ — B.
P..hSjr
The matrix B is a (MxM) diagonal matrix in which each (m,m) diagonal term
consists of B,„, is the transpose matrix of C, the (NxK) matrix Df
determines coupling between the K point force locations and the M structural
modes, gp is the generalised modal force vector due to the primary plane wave
excitation, the K length vector fc is the complex vector of structural control
point forces, and a is the complex acoustic modal amplitude vector. The
(MxM) diagonal matrix Ys can be defined as the uncoupled structural modal
mobility matrix which determines the relationship between structural
excitation and the resultant structural velocity response in modal co-ordinates
of the uncoupled structural system. As with the uncoupled acoustic impedance
matrix Za, note that Ys is a diagonal matrix.
From (Eq. 14) and (Eq. 19), we get
a = (l + Z,CY,C’^)''z,(D,q,+CY,g,+CWc)
(20)
b = (I + Y,C’'Z,C)'' Y,(gp + Dff, - C^Z.D^q,)
(21)
Since weakly coupling is assumed i.e. = 0 and Y^C^Z^C = 0 ,
then
we get
a = Z,(D,q,+CY,g,+CY,D,f.)
(22)
b = X(g,+D,f,-C%D,q,)
(23)
Although the formulation developed above covers fully coupled systems, weak
coupling is assumed hereafter for the convenience of analysis.
1238
In order to minimise the sound transmission into the cavity, two kinds of
actuators are used: a single point force actuator for controlling the structural
vibration of the plate and a single rectangular type acoustic piston source for
controlling the cavity acoustic pressure. The rectangular piston source is
centred at (1.85,0.15,0) with the area of 0.15m by 0.15m, This location was
chosen because the sound pressure of each mode in a rectangular cavity is a
maximum at the corners, and thus the control source is placed away from the
acoustic nodal planes [2]. For a similar reason, the point force actuator is
located at (9/20Li, L2/2) on the plate, at which there are no nodal lines within
the frequency range of interest. Table 2 shows the natural frequencies of each
uncoupled systems and their geometric mode shape coupling coefficients
which are normalised by their maximum value. Some of natural frequencies
which are not excited by the given incident angie((p = 0°) were omitted. The
(m/, m2) and («/, 112, ns) indicate the indices of the m-th plate mode and the n\h
cavity mode, and corresponding the uncoupled natural frequencies of the plate
and the cavity are listed. A total 15 structural and 10 acoustic modes were used
for the analysis under 300 Hz, and no significant difference was noticed in
simulations with more modes.
3.2 Active minimisation of the acoustic potential energy
This section considers an analytical investigation into the active control of the
sound transmission into the rectangular enclosure in Figure 2. Three active
control strategies classified by the type of actuators are considered. They are;
i) a single force actuator, ii) a single acoustic piston source, and iii)
simultaneous use of both the force actuator and the acoustic piston source.
Although the formulation developed in this paper is not restricted to a single
actuator, each single actuator was used to simplify problems so that the control
mechanisms could be understood and effective guidelines for practical
implementation could be established.
3.2.1 control using a single force actuator
A point force actuator indicated in Figure 2 is used as a structural actuator and
the optimal control strength of the point force actuator can be calculated using
(Eq. 26). Figure 3(a) shows the acoustic potential energy of the cavity with
and without the control force. To show how this control system affects the
vibration of the plate, the vibration kinetic energy of the plate obtained from
(Eq. 8) is also plotted in Figure 3(b). On each graph, natural frequencies of the
plate and the cavity are marked and ‘o’ at the frequencies, respectively. It
can be seen that the acoustic response of uncontrolled state has peaks at both
1239
plate and cavity resonances, and the vibration response of uncontrolled state is
governed by the plate resonances only because of ‘weak coupling’.
Examining Figure 3(a,b) it can be seen that at the 1st, 2nd, 4th, and 5th plate
modes corresponding to 52 Hz, 64 Hz, 115 Hz, and 154 Hz, respectively there
is a large reduction of the acoustic potential energy. This is because the sound
field at these frequencies is governed by the plate vibration modes, and a
single structural actuator is able to control the corresponding vibration mode to
minimise sound transmission.
The structural actuator reduces sound at cavity-controlled modes as well(
especially the 2nd and 3rd cavity modes corresponding respectively to 85 Hz
and 170 Hz), however it has to increase plate vibration significantly. It shows
that minimisation of the acoustic potential energy does not always bring the
reduction of structural vibration, and vice versa. Since a cavity-controlled
mode is generally well coupled with several structural modes, a single
structural actuator is not able to deal with several vibration modes because of
'control spillover\A\. This is the reason why a single acoustic piston source
used in the next section was introduced. However, it is clear that a single
point force actuator is effective in controlling a well separated plate-controlled
mode provided the actuator is not located close to the nodal line.
3.2.2 control using a single piston force source
A single acoustic piston source indicated in Figure 2 is used for controlling
the acoustic sound field directly. The optimal control source strength of the
acoustic piston source can be determined using (Eq. 26). Figure 4 shows the
acoustic potential energy of the cavity and the vibrational kinetic energy of the
plate with and without the control actuators.
Since a plate-controlled mode is generally coupled with several cavity modes,
the control effect of the acoustic source is not effective at plate-controlled
modes (e.g. 52 Hz, 64 Hz, 115 Hz etc.). Whereas, it is more able to reduce
transmitted sound at the cavity-controlled modes ( e.g. 0 Hz, 85 Hz, 170 Hz
and 189 Hz) than the structural actuator. As can be noticed from (Eq. 22), the
external incident wave and force excitation have the same sound transmission
mechanism, which is vibrating the plate and transmitting sound through the
geometric mode shape coupling matrix C. Thus, it can be said that the
structural actuator is generally effective in controlling sound transmission. At
cavity controlled modes, however, several vibration modes are coupled with
an acoustic mode. It means that a single acoustic source is more effective than
a single structural actuator since a single actuator is generally able to control
only one mode. From the results, it is clear that a single acoustic piston source
is effective in controlling well separated cavity-controlled modes. It is
interesting that there is not much difference in the vibrational kinetic energy
1240
with and without control state. It means that the acoustic actuator is able to
reduce sound field globally without increasing plate vibration.
3.2.3 control using both the piston source and the structural actuator
In this section, a hybrid approach, simultaneous use of both the point force
actuator and the acoustic piston source, is applied. The optimal strength of the
force actuator and the piston source can be obtained from Eq. (26). Figure 5
shows the acoustic potential energy of the cavity and the vibrational kinetic
energy of the plate with and without the control actuators. Even at the cavity-
controlled modes, it can be seen that a large reduction in the acoustic potential
energy is achieved without significantly increasing the structural vibration. In
general, more control actuators result in better control effects[6]. However, the
hybrid approach with both structural and acoustic actuators in the system does
not merely mean an increase in the number of actuators. As demonstrated in
the last two sections, a single structural actuator is effective in controlling
well separated plate-controlled modes and a single acoustic actuator is
effective for controlling well separated cavity-controlled modes. Since the
acoustic response is governed by both plate-controlled and cavity-controlled
modes, the hybrid control approach can be desirable for controlling sound
transmission in a ‘weakly-coupled’ structural acoustic system.
To investigate the control efforts of each control system, the amplitude of the
force actuator and the source strength of the piston source are plotted in
Figure 6. There is a large decrease of the force amplitude at the well
separated cavity-controlled modes, e.g. 85 Hz and 170 Hz, by using the both
actuators. This trend can also be seen in the case of the piston source strength,
especially at the 1st and 2nd structural natural frequency (52 Hz, 64 Hz). By
using the hybrid approach, simultaneous use of both actuators, better control
effects on the vibration of the plate, the transmission noise reduction and the
control efforts of the actuators can be achieved.
4. Conclusion
The active control of the sound transmission into a ‘weakly coupled’
structural-acoustic system has been considered. The results obtained
demonstrates that a single point force actuator is effective in controlling well
separated plate-controlled modes, whereas, a single acoustic piston source is
effective in controlling well separated cavity-controlled modes.
By using the hybrid approach with both structural and acoustic actuators,
improved control effects on the plate vibration, further reduction in sound
1241
transmission, and reduced control efforts of the actuators can be achieved.
Since the acoustic behaviour is governed by both plate and cavity resonances,
the hybrid control approach can be desirable in controlling sound transmission
in a ‘weakly coupled’ structural-acoustic system.
References
1. C.R. FULLER, S.J. ELLIOTT and P.A. NELSONActive control of
vibration, Academic Press Limited, 1996
2. P.A. NELSON and S.J. ELLIOTT Active control of sound, Academic
Press Limited, 1992
3. B.-T. WANG, C.R. FULLER and K. DIMITRIADIS Active control of
noise transmission through rectangular plates using multiple
piezoelectric or point force actuators Journal of the Acoustical Society
of America, 1991, 90(5), 2820-2830.
4. M.E. JOHNSON and S. J. ELLIOTT Active control of sound radiation
using volume velocity cancellation. Journal of the Acoustical Society of
America, 1995, 98(4), 2174-2186.
5. C.R. FULLER, C.H. HANSEN and S.D. SNYDER Active control of
sound radiation from a vibrating rectangular panel by sound sources
and vibration inputs: an experimental comparison, Journal of Sound
and Vibration, 1991, 145(2), 195-215.
6. L. MEIROVrrCH and S. THANGJITHAM Active control of sound
radiation pressure, Trans, of the ASMS Journal of Vibration and
Acoustics, 1990, 112, 237-244.
7. S.D. SNYDER and N. TANAKA On feedforward active control of
sound and vibration using vibration error signals, Journal of the
Acoustical Society of America, 1993, 94(4), 2181-2193.
8. J. PAN C.H. HANSEN and D. A. BIES Active control of noise
transmission through a panel into a cavity : I. analytical study. Journal
of the Acoustical Society of America, 1990, 87(5), 2098-2108.
9. P.M. MORSE and K.U. INGARD Theoretical Acoustics, McGraw-
Hill, 1968
10. E.H. DOWELL, G.F. GORMAN HI, and D.A. SMITH
Acoustoelasticity : general theory, acoustic modes and forced response
to sinusoidal excitation, including comparisons with experiment.
Journal of Sound and Vibration, 1977, 52(4), 519-542.
11. F. FAHY Sound and Structural Vibration, Radiation, Transmission
and Response, Academic Press Limited, 1985
1242
incident plane wave
Figure 1 A structural acoustic coupled system with the volume V and its flexible boundary
surface S/.
Figure 2 The rectangular enclosure with one simply supported plate on the surface Sf on
which external plane wave is incident with the angles of (cp = 0°) and (0=45°).
Table 1 Material properties
Material
Density
Phase speed
Young’s
Poisson’s
Damping
(kg/m^)
(m/s)
modulus (N/m^)
ratio (v)
ratio (0
Air
1.21
340
-
-
0.01
Steel
7870
-
207x10®
0.292
0.01
1243
Table 2 The natural frequencies and geometric mode shape coupling coefficients of each
uncoupled system
Order:
Plate,
1
2
3
4
■„ , 5.
7
10
■c
’Type
(1,1)
(2,1)::
0,1) :
(4,1)
(5,1)
(6,1)
(7,1)
Cavity^
' Freq.-
52 Hz
64 Hz
86Hz
115 Hz
154 Hz
200 Hz
;256Hz
1
(0,0,0)
0 Hz
0.71
0
0.24
0
0.14
0
O.IO
2
(1,0.0)
85 Hz
0
0.67
0
0.27
0
0.17
0
3
(2,0.0)
170 Hz
-0.33
0
0.60
0
0.24"
0
0.16
4
(0,0.1)
189 Hz
-1.00
0
-0.33
0
-0.20"
0
-0.14
5
(1,0,1)
207.HZ
0
-0.94
0
-0.38
0
-0.24
0
6
(2;0,1)
254 Hz
0.47
0
-0.85
0
-0.34
0
-0.22
7
(3,0,0)
255 Hz
0
-0.40
0
0.57
0
0.22
0
1 oo ISO 200 2SO
Frequency
(a) the acoustic potential energy of the cavity
TSO
Frequency
(b) the vibrational kinetic energy of the plate(dB ref .= 10 ^ J)
Figure 3 Effects of minimising the acoustic potential energy using a point force actuator (
solid line : without control, dashed line : with control ), where **’ and ‘o’ are at uncoupled
plate and cavity natural frequencies, respectively.
301
Frequency
(a) the acoustic potential energy of the cavity
Figure 4 Effects of minimising the acoustic potential energy using an acoustic piston source
( solid line ; without control, dashed line : with control ), where and ‘o’ are at uncoupled
plate and cavity natural frequencies, respectively.
Figure 5 Effects of minimising the acoustic potential energy using both a point force
actuator and an acoustic piston source - continued
2
Frequency
(b) the vibrational kinetic energy of the pIate(dB ref =10‘^ J)
Figure 5 Effects of minimising the acoustic potential energy using both a point force
actuator and an acoustic piston source ( solid line : without control, dashed line: with control
), where and ‘o’ are at uncoupled plate and cavity natural frequencies, respectively.
Frequency
(a) the strength of the force actuator
Frequency
(b) the strength of the piston source( unit: mVsec)
Figure 6 Comparison of control efforts of the three control strategies; using each actuator
separately ( solid line ) and using both the force actuator and the piston source (dashed line )
, where and ‘o’ are at uncoupled plate and cavity natural frequencies, respectively.
1246
A DISTRIBUTED ACTUATOR FOR THE
ACTIVE CONTROL OF SOUND
TRANSMISSION THROUGH A PARTITION
TJ. Sutton, M.E. Johnson and S.J. Elliott
Institute of Sound and Vibration Research
University of Southampton, Southampton S017 IBJ
ABSTRACT
The paper considers the problem of active control of soimd transmission
through a partition using a single distributed actuator. The use of shaped,
distributed actuators rather than point sources or locally-acting piezoceramic
elements offers the possibility of controlling the volume velocity of a plate
without giving rise to control spillover and avoids an increase in the sound
radiated by uncontrolled structural modes. Specifically, a form of distributed
piezoelectric actuator is described in which the electrode takes the form of a
set of quadratic strips and serves to apply a roughly uniform normal force
over its surface.
INTRODUCTION
The strong piezoelectric properties of the polymer polyvinylidene fluoride
(PVDF) were discovered in 1969 [1]. The material is lightweight, flexible,
inexpensive and can be integrated into engineering structures for strain
sensing and to apply distributed forces and moments for the active control of
vibration and sound transmission. Such ^smarf materials offer the possibility
of providing lightweight sound-insulating barriers for application to aircraft,
ground-based transport and in buildings.
Lee [2] has set out the underlying theory of active laminated structures in
which one or more layers of flexible piezoelectric material are attached to a
plate. Practical sensors using PVDF material have been implemented by
Clark and Fuller [3], Johnson and Elliott [4-6], and others. In these cases thin
PVDF films were attached to the structure to sense integrated strain over a
defined area. In [4] for example a distributed sensor was developed whose
output is proportional to the integrated volume velocity over the surface of a
plate.
1247
A number of studies have been carried out in which distributed piezoelectric
actuators form a layer of a laminated system. In [7] the shape of a distributed
piezoelectric actuator was chosen to be orthogonal to all but one of the natural
modeshapes of the cylindrical shell system xmder control. Using this
approach a set of actuators could be matched to the modes of the system
under control, avoiding control spillover {i.e. the excitation of tmcontrolled
structural modes).
In the present paper a single shaped PVDF actuator is applied to a thin plate
to control the noise transmission through it. The shape of the actuator is
chosen specifically to apply an approximately uniform force to the plate.
Such an actuator can be used to cancel the total volume velocity of the plate
and therefore substantially to reduce the radiated sound power. (If volume
velocity is measured at the plate surface there is no requirement for a remote
error microphone.) As noted by Johnson and Elliott [6], the soimd power
radiated by a plate which is small compared with an acoustic wavelength
depends mainly on the volume velocity of the plate. The simulations in [6]
show that provided the plate is no larger than half an acoustic wavelength, a
single actuator used to cancel volume velocity will achieve similar results to a
strategy in which radiated power is minimised. It is possible to envisage a
large partition made up of a number of active plate elements designed on this
basis.
CALCULATION OF NORMAL FORCES IN THE PLATE
In this section the equation of motion of the plate and attached PVDF layers is
set out. The film thickness is assumed to be 0.5 mm. The analysis broadly
follows that of Dimitriadis, Fuller and Rogers [8], but the individual
piezoelectric coefficients and included separately as is appropriate
for PVDF and a sensitivity function is included to account for variations in
electrode shape. In addition the bending stiffness of the piezoelectric film is
included (it is not negligible as the whole plate is covered). The nomenclature
matches that used by Fuller, Elliott and Nelson [9] but here the analysis leads
to the inhomogeneous wave equation for the plate-actuator system.
We consider an aluminium plate of thickness Ih^ as shown in Figure 1. The
plate is covered on its upper and lower surfaces with a piezoelectric film of
sensitivity (}>(x,y)d-p in which d.^ is the strain/ electric field matrix of the material
(3x6 array) and <^{x,y) is a spatially-varying sensitivity function
(0 < (j)(x,y) < 1 ). The two piezoelectric films are assumed to be identical but
the same drive voltage is applied with opposite polarity to the lower film. As
a result of this antisymmetric arrangement, the plate is subject to pure
bending with no straining of the plate midplane.
1248
In line with other similar calculations [2,9] it is assumed that any line
perpendicular to the midplane before deformation will remain perpendicular
to it when the plate /PVDF assembly is deformed. As a result, the strain at
any point in the assembly is proportional to distance z through it. (z=0 is
defined to be on the midplane of the plate as indicated in Figure 1.) The
direct and shear strains throughout the whole assembly (e^, Ey, E^y) are then
given by [10]:
3^^
d^w
(1-3)
in which w is the displacement of the midplane in the z direction.
The corresponding stresses in the plate (only) follow from Hooke's law as in
the standard development for thin plates:
p _ _ p_
=
i-v:
A ^ p
0'' = — — -
2a+vj
(4-6)
in which is the Young's modulus and Vp is Poisson's ratio for the plate
material. Stress in the piezoelectric film follows from the constitutive
equations for the material [2]. The direct and shear stresses for the upper
piezo film are designated of of and respectively:
's/
K
or"’
= C ^
1 -
CO
_^36.
in which is the voltage applied across the actuator fibn (thickness /rj and
4 and are the strain/field coefficients for the material. For PVDF d^, =
0, but it is included in the analysis for completeness. As in [2], the stiffness
matrix C is given by:
1249
0
c
^pe ^ pe^pe
1 —
^ ^ pe ^ pe
^ pe^pe ^pe
1-v' 1-v^
0 0
0
^pe
2(l + v^J
(8)
The stresses in the lower piezoelectric layer are designated and
, and the form of the expression is similar to the upper layer except that
the voltage is applied with reversed polarity:
's/
^31
V
= c
+ (^{x,y)
a
Summing moments about the x and y axes for a small element dxby of the
plate yields the moment per unit length about the y-axis and My about
the x-axis; also the corresponding twisting moments per imit length, My^ and
M^=jc!’’zdz+
-III, ~K-K ^‘b
(10)
h,, -III, ‘‘b+lh,
My = j a^yZdz + J + J cf'^zdz
-III, -hh-K h
(11)
h,, -III, l>h+lh,
= |<t;,z*+ ja^fhdz
(12)
-III, -K-K h
and My^=M,y.
The vertical acceleration at each point of the plate d^w/dt^ is obtained by
taking moments about the x and y axes for a small element 6x6y and
resolving vertical forces as in standard thin plate theory. If the plate is acted
on by some external force per unit area p(x,y)f(t) then the vertical motion of
the plate is described by:
dx^ dxdy dy^ dt^
= -p(x,y)f(t)
(13)
1250
in which m is the mass/ area of the plate-film assembly.
The equation of motion of the plate complete with attached piezoelectric film
is obtained by combining the above equations. For convenience the following
constants are defined;
n _
'' 3(1
(bending stiffness of plate, thickness 2/i,) (14)
3(1 -V^)
(bending stiffness due to PVDF) (15)
+2/1,)
The equation of motion of the plate assembly including upper and lower
piezoelectric layers is then given by:
3^(l)(^,y)
3>(j:,y)
The left-hand side of this equation determines the free response of the plate-
film assembly and is recognised as the standard form for a thin plate. The
first term on the right-hand side is the assumed externally applied normal
force per imit area. The second term on the right-hand side gives the effective
normal force per unit area applied to the plate due to the two piezoelectric
films driven by a voltage . It is clear that this force depends on the
spatially-varying sensitivity which has been assumed for the piezoelectric
material.
An examination of Eqns. (16) and (17) shows that the normal force applied by
the piezoelectric film depends on the sum of the plate thickness and the
thickness of one of the film layers. If the film is much thinner than the plate
(h^ « hy) then the applied force becomes independent of the film thickness and
depends only on the plate thickness, the applied voltage, the electrode pattern
and the material constants. Eqn. (17) also shows that the normal force is
applied locally at all points on the plate. No integration is involved, and so in
contrast with a volume velocity sensor designed using quadratic strips [5], the
force does not depend on the plate boundary conditions in any way.
Furthermore, there is no need to use two films oriented at 90 degrees to cancel
1251
the cross-sensitivity It is also worth noting that no assumption has been
made about the modeshapes on the plate.
We can create a uniform force actuator by choosing:
^ = constant, and (18)
Bx
(19)
This can approximately be achieved by depositing electrodes in the form of
narrow strips whose width varies quadratically in the x-direction. (Note that
the x-direction is defined as the direction of rolling of the PVDF material, Le.
the direction of for maximum sensitivity.) The form of the electrodes is as
shown in Figure 2. With this pattern the sensitivity function takes the form:
(t)(x,>') = (20)
where is the length of the strip. Thus (t)(-^>^) - ^ at x = 0 and x = (no
electrode), while (|)(x, 31) = 1 halfway along at x = LJ2 (electrode fully covers
the film).
APPLICATION TO A THIN ALUMINIUM PLATE
If a plane wave of sound pressure level 94 dB (say) is normally incident on a
hard surface, it will exert a pressure of 2 Pa rms on that surface. If this
incident pressure is counterbalanced by a uniform force actuator applied to a
plate, then the plate could in principle be brought to rest. Thus for active
control of everyday noise levels the uniform force actuator will need to be
able to generate a normal force /area of a few pascal over the surface of the
plate. (When the incident wave impinges on the plate at an oblique angle,
many natural modes of the plate will be excited and it will not be possible to
bring it perfectly to rest with a single actuator; however it will remain
possible to cancel the plate volume velocity as explained earlier.)
By way of example an aluminium plate of thickness 1 mm will be assumed,
with a free surface measuring 300 x 400 mm. Attached to each side is a PVDF
film of thickness 0.5 mm. One electrode of each panel would be masked to
give quadratic strips of length 300 mm as shown in Figure 2. (The width of the
strips is unimportant, but should be significantly smaller than the structural
wavelength of modes of interest on the plate.) In this case it turns out that
= 6.64 for the plate, and
1252
Dp, = 1.27 for the PVDF film.
The piezoelectric constants for the film are typically
4 = 23 X m/V and
4 = 3 X lO’"' m/V, giving
Cp, = 3.28x10^
Finally the double derivative of the sensitivity function turns out to be
= 88.9 (300 mm strip length)
dx^
The bending stiffness of the 1 mm aluminium plate is increased by 20% due to
the addition of two layers of PVDF film of thickness 0.5 mm each. The force
per unit area due to the actuator is obtained from Eqn. (17):
force/area = 4,(^3, +Vp,4)|^'^3
= 6.96x10’^ V3 Pa
Thus 1000 volt rms would yield a tmiform force/area of close to 7 Pa. This is
not an impracticable voltage level, but previous experience at ISVR suggests
that care would need to be taken to avoid electrical breakdown through the
air between electrodes, or over damp surfaces.
ACTIVE CONTROL OF SOUND TRANSMISSION
In reference [6] Johnson and Elliott have presented simulations of the active
control of harmonic sound transmitted through a plate using a uniform force
actuator. Their actuator might be realised along the lines described in this
paper. In the simulatioi\s presented in [6] the uniform force actuator is used
with a matched volume velocity sensor having the same electrode shape [4].
The advantage of this configuration is that the actuator can be used to drive
the net volume velocity of the plate to zero without exciting high order
structural modes in the process (control spillover). Simulations of a
300x380x1 mm aluminium plate showed that reductions in transmitted sound
power of around 10 dB were achievable in principle up to 600 Hz using this
matched actuator-sensor arrangement.
A further advantage of the distributed matched actuator-sensor pair is that
the secondary path through the plate (for active control) is minimum phase
[6], giving good stability characteristics if a feedback control loop is
implemented to control random incident soimd for which no reference signal
is available.
1253
CONCLUSION
A design of distributed piezoelectric actuator has been presented which
generates a roughly uniform force over the surface of a plate. An example
calculation shows that the design is capable of controlling realistic soimd
pressure levels. When used in combination with a matched volume velocity
sensor, the actuator-sensor pair will have minimum-phase characteristics and
will offer the possiblity of feedback control in which neither a reference signal
nor a remote error sensor will be required.
ACKNOWLEDGEMENT
The financial support of the European Community under the Framework IV
programme is gratefully acknowledged. (Project reference: BRPR-CT96-0154)
REFERENCES
[1] G.M. Sessler (1981) JAcoust Soc Am 70(6) Dec 1981 1596-1608
Piezoelectricity in polyvinylidene fluoride
[2] C.K. Lee (1990) JAcoust Soc Am 87(3) Mar 1990 1144-1158 Theory of
laminated piezoelectric plates for the design of distributed sensors/ actuators.
Part I: Governing equations and reciprocal relationships
[3] R.L. Clark and C.R. Fuller (1992) JAcoust Soc Am 91(6) June 1992 3321-3329
Modal sensing of efficient acoustic radiators with polyvinylidene fluoride
distributed sensors in active structural acoustic control approaches
[4] M.E. Johnson, S.J. Elliott and J.A. Rex (1993) ISVK Technical Memorandum
723. Volume Velocity Sensors for Active Control of Acoustic Radiation
[5] M.E. Johnson and S.J. Elliott (1995) Proceedings of the Conference on Smart
Structures and Materials 27 Feb-3 Mar 1995, San Diego, Calif. SPIE Vol 2443.
Experiments on the active control of sound radiation using a volume velocity
sensor
[6] M.E. Johnson and S.J. Elliott (1995) JAcoust Soc Am 98(4) Oct 1995 2174-
2186. Active control of sound radiation using volume velocity cancellation
[7] H.S. Tzou, J.P. Zhong and J.J. Hollkamp (1994) Journal of Sound and
Vibration 177(3) 363-378 Spatially distributed orthogonal piezoelectric shell
actuators: theory and applications
[8] E.K. Dimitriadis, C.R. Fuller and C.A. Rogers (1991) Transactions of the
ASME, Journal of Vibration and Acoustics 113 100-107 Piezoelectric actuators for
distributed vibration excitation of thin plates
[9] C.R. Fuller, S.J. Elliott and P.A. Nelson (1996) Active Control of Vibration.
Academic Press, London.
[10] G.B. Warburton (1976) The Dynamical Behaviour of Structures, 2nd Edition.
Pergamon Press, Oxford.
1254
layers of
PVDF film
Figure 1 : Schematic diagram of thin plate
covered on both sides with a layer of PVDF film
Figure 2: Electrode pattern of quadratic strips for
uniform-force actuator
1255
1256
CONTROL OF SOUND RADIATION FROM A FLUID-LOADED PLATE
USING ACTIVE CONSTRAINING LAYER DAMPING
J. Ro, A. Al-Ali and A. Baz
Mechanical Engineering Department
The Catholic University of America
Washington D. C. 20064
Abstract
Sound radiation from a vibrating flat plate, with one side subjected to
fluid-loading, is controlled using patches of Active Constrained Layer Damping
(ACLD). The fluid-structure-controller interaction is modeled using the finite
element method. The damping characteristics of the ACLD/plate/fluid system are
determined and compared with the damping characteristics of plate/fluid system
controlled with conventional Active Control (AC) and/or Passive Constrained
Layer Damping (PCLD) treatments. Such comparisons are essential in
quantifying the individual contribution of the active and passive damping
components to the overall damping characteristics, when each operates separately
and when both are combined to interact in unison as in the ACLD treatments.
I. INTRODUCTION
When a structure is in contact with or immersed in a fluid, its vibration
energy radiates into the fluid domain. As a result, there is an observable increase
in the kinetic energy of the structure due to the fluid loading. Because of this
kinetic energy increase, the natural frequencies of structures which are subjected
to fluid-loading decrease significantly compared to the natural frequencies of
structures in vacuo. Therefore, through understanding of the interaction between
the elastic plate structures and the fluid loading has been essential to the effective
design of complex structures like ships and submarine hulls. Lindholm et al. [1]
used a chordwise hydrodynamic strip theory approach to study the added mass
factor for cantilever rectangular plates vibrating in still water. Fu et al [2] studied
the dry and wet dynamic characteristics of vertical and horizontal cantilever
square plates immersed in fluid using linear hydroelasticity theory. Ettouney et
al [3] studied the dynamics of submerged structures using expansion vectors,
called wet modes which are finite series of complex eigenvectors of the fluid-
structure system. Recently Kwak [4] presented an approximate formula to
estimate the natural frequencies in water from the natural frequencies in vacuo.
When the structure and the fluid domains become rather complex,
solutions of fluid-structure coupled system can be obtained by finite element
1257
methods. Marcus [5], Chowdhury [6], Muthuveerappan et al. [7] and Rao et al
[8] have successfully implemented the finite element method to predict the
dynamic characteristics of elastic plates in water. Everstine [9] used both finite
and boundary element methods to calculate the added mass matrices of fiilly-
coupled fluid-structure systems.
The above investigations formed the bases necessary to devising passive
and active means for controlling the vibration of as well as the sound radiation
from fluid-loaded plates. Passive Constrained Layer Damping (PCLD) treatments
have been used extensively and have proven to be effective in suppressing
structural vibration as reported, for example, by Jones and Salerno [10], Sandman
[11] and Dubbelday [12]. Recently, Gu and Fuller [13] used feed-forward control
algorithm which relied in its operation on point forces to actively control the
sound radiation from a simply-supported rectangular fluid-loaded plate.
In the present study, the new class of Active Constrained Layer Damping
(ACLD) treatment is utilized as a viable alternative to the conventional PCLD
treatment and Active Constrained (AC) with PCLD treatment (AC/PCLD). The
ACLD treatment proposed combines the attractive attributes of both active and
passive damping in order to provide high energy dissipation-to-weight
characteristics as compared to the PCLD treatment. Such surface treatment has
been successfully employed to control the vibration of various structural members
as reported, for example, by Shen [14] and Baz and Ro [15]. In this paper, the use
of the ACLD is extended to the control of sound radiation from fluid-loaded
plates. Finite element modeling of the dynamics and sound radiation of fluid-
loaded plates is developed and validated experimentally. Particular focus is
placed on demonstrating the effectiveness of the ACLD treatment in suppressing
the structural vibration and attenuating the sound radiation as compared to
conventional PCLD and AC/PCLD.
This paper is organized in five sections. In Section 1, a brief introduction
is given. In Section 2., the concepts of the PCLD, ACLD and AC/PCLD
treatments are presented. In Section 3, the dynamical and fluid finite element
models are developed to describe the interaction between the plate, ACLD and the
contacting fluid. Experimental validation of the models are given in Section 4.
Comparisons between the theoretical and experimental performance are also
presented in Section 4 for different active and passive damping treatments.
Section 5, summarizes the conclusions of the present study.
2. CONCEPTS OF PCLD, ACLD AND AC/PCLD TREATMENTS
Figures (1-a), (1-b) and (1-c) show schematic drawings of the PCLD,
ACLD and AC/PCLD treatments respectively. In Figure (1-a), the plate is treated
1258
Figure (1) - Schematic drawing of different surface treatments (a) PCLD, (b)
ACLD and (c) AC/PCLD.
with a viscoelastic layer which is bonded directly to the plate. The outer surface
of the viscoelastic layer is constrained by an inactive piezo-electric layer in order
to generate shear strain y, which results in dissipation of the vibrational energy of
the plate. Activating the constraining layer electrically, generates a control force
Fp by virtue of the piezo-electric effect as shown in Figure (l-b) for the ACLD
treatment . Such control action increases the shear strain to yj which in turn
enhances the energy dissipation characteristics of the treatment. Also, a restoring
moment Mp=d2Fp is developed which attempts bring the plate back to its
undeformed position. In the case of AC/PCLD treatment, shown in Figure (l-c),
two piezo- films are used. One film is active and is bonded directly to the plate to
control its vibration by generating active control (AC) force Fp and moment
Mp^djFp. The other film is inactive and used to restrain the motion of the
1259
viscoelastic layer in a manner similar to the PCLD treatment of Figure (1-a). In
this way, the AC action operates separately from the PCLD action. This is unlike
the ACLD configuration where the active and passive control actions operate in
unison. Note that in the ACLD configuration, larger shear strains are obtained
hence larger energy dissipation is achieved. Furthermore, larger restoring
moments are generated in the ACLD treatments as compared to the AC/PCLD
treatments as the moment arm d2 in the former case is larger than the moment arm
d3 of the latter case. This results in effective damping of the structural vibrations
and consequently effective attenuation of sound radiation can be obtained.
3. FINITE ELEMENT MODELING
3.1 Overview
A finite element model is presented in this section, to describe the
behavior of fluid-loaded thin plates which are treated with ACLD, PCLD and
AC/PCLD treatments.
3.2 Finite Element Model of Treated Plates
(b) (c)
Figure (2) - Schematic drawing of plate with ACLD/AC/PCLD patches.
Figure (2) shows a schematic drawing of the ACLD and AC/PCLD
treatments of the sandwiched plate which is divided into N finite elements. It is
assumed that the shear strains in the piezo-electric layers and in the base plate are
negligible. The transverse displacement w of all points on any cross section of the
sandwiched plate are considered to be the same. The damping layers are assumed
to be linearly viscoelastic with their constitutive equations described by the
complex shear modulus approach such that G=G (1+T|i). In addition, the bottom
piezo-electric layer (AC) and the base plate are considered to be perfectly bonded
together and so are the viscoelastic layer and the top piezo-electric layer.
The treated plate elements considered are two-dimensional elements
bounded by four nodal points. Each node has seven degrees of fireedom to
describe the longitudinal displacements u, and v, of the constraining layer, U3 and
V3 of the base plate, the transverse displacement w and the slopes w and W y of
the deflection line. The deflection vector {5} can be written as:
{5} = {u„v„U3, V3,W, W ^W y}""
= [{n,} {N3} {Nj {n.} {n,} {N,}^
where {5"} is the nodal deflection vector, {Nj}, {Nj}, {N3}, {N4}, {N5}, {N5} ,,,
and {N5} y are the spatial interpolating vectors corresponding to u„ v„ U3, V3, w,
w^, and Wy respectively. Subscripts ,x and ,y denote spatial derivatives with
respect to x and y.
Consider the following energy functional ITp for the treated plate/fluid
system:
np = I(u-TK+w.-w^+wJdv, (2)
where U is the strain energy, T^ is the kinetic energy, is work done by external
forces, Wp is work done by the back pressure inside the fluid domain, is work
done by the control forces and moments and V is the volume of the plate. These
energies are expressed as follows
I T,dv =X ^ 1 1„ (*' = 5{®T['^p]{®') ’
|w,dV=j{5'f{F},
= 5' Kj 5'
“ V.l^dxdy
„[Ui hi ‘J “ ax^ Ui ‘ hi j “ Sy^J
where i=l for ACLD control or i=3 for AC control (6)
and I WpdV ={5'}’"[n ]{p'}.
where {p®} is the nodal pressure vector of the fluid element. In the above
equation []^], [Mp], {F}, [KJ and [Q] are the plate stiffness matrix, mass matrix,
external forces vector, piezo-electric forces and moments matrix and plate/fluid
coupling matrix as given in the appendix. In equation (6), d3i 32 are the piezo¬
strain constants in directions 1 and 2 due to voltage applied in direction 3. The
voltage is generated by feeding back the derivative of the displacement 5 at
critical nodes such that j where is the derivative feedback gain
matrix and C is the measurement matrix defining the location of sensors.
Minimizing the plate energy fimctional using classical variational methods
such that |anp/a{6®}j = 0 leads to the following finite element equation:
{[K]-»lM.]){5'}-[n]{p'} = {F} (8)
where co is the frequency and [K] = [Kp] + [K^.] is overall stiffriess matrix.
3.3 Finite Element Model of the Fluid
The fluid model uses solid rectangular tri-linear elements to calculate the
sound pressure distribution inside the fluid domain and the associated structural
coupling effects. The fluid domain is divided into fluid elements. Each of
1262
these elements has eight nodes with one degree of jfreedom per node. The
pressure vector is expressed by p = [Nf]{p®} and [NJ is pressure shape function
and {p®} is nodal pressure vector.
Considering the following functional Elf of fluid domain Craggs [16]
where [Kf] and [MJ are the fluid stiffiiess and mass matrices as given in the
appendix. Minimization of equation (9) such that {OTf/^lp^}} = 0 yields the
fluid dynamics as coupled with the structural vibration:
([K,]-o)^[M,]){p'}-<a^[nf{5'} = {0} (10)
The boundary conditions involved are of the form
ap/an = 0, at a rigid boundary
a p / a n = -pp 9^5 / 9 ,
and p = 0.
at a vibrating boundary
at a free surface
where pf is fluid density.
3.4. Solutions of the Coupled Plate/Fluid System
Combining equations (8) and (10) gives
■[K]-co^[m,] -[fi] Ip'l.pl (11)
[Kr]-CD^[M,]J Lp'J k
At low frequencies, the fluid pressure is in phase with the structural
acceleration, i.e. the fluid appears to the structure like an added mass. However,
as the frequency increases the added mass effect diminishes and the damping
effect, i.e. the pressure proportional to velocity, increases. For an incompressible
fluid, the speed of sound c approaches oo, thus the mass matrix of the fluid [MJ
vanishes, and equation (11) can be simplified to
■[K]-cd=[m,] -[fl]] rs'] pi (12)
_ [KfiJkJ k
If the fluid-structure coupled system has free boundary surface, then [KJ is
non-singular [Everstin, 1991] and the nodal pressure vector {p®} can be eliminated
from equation (12) as follows:
{p'}=-o.^[K,nnr{5'} (13)
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Hence, equation (12) yields
([K]-ffl^(K]+[Mj)){8') = {F) (14)
where [MJ is added virtual mass matrix defined by [Muthuveerappan, 1979]
[Mj = [a][K,]-'[£2f (15)
Equation (14) only involves the unknown nodal deflection vector {5®} of the
structure. When {F}=0, equation (14) becomes an eigenvalue problem, the
solution of which yields the eigenvalues and eigenvectors. The nodal pressures
can then be obtained from equation (13) when the nodal displacements are
determined for any particular loading on plate.
4. PERFORMANCE OF PARTIALLY TREATED PLATES WITH
FLUID LOADING
In this section, comparisons are presented between the numerical
predictions and experimental results of the natural frequencies and damping ratios
of a fluid-loaded plate treated with ACLD, PCLD and AC/PCLD. The effect of
the Active Control, Passive Constrained Layer Damping and Active Constrained
Layer Damping on the resonant frequency, damping ratio, attenuation of vibration
amplitude and sound radiation are investigated experimentally. The vibration and
sound radiation attenuation characteristics of the fluid-structure coupled system
are determined when the plate is excited acoustically with broadband frequency
excitation while the piezo-electric layers are controlled with various control gains.
The experimental results are compared with the theoretical predictions.
4.1 Experimental Set-up
Figures (3-a) and (3-b) show a schematic drawing and finite element mesh
of the experimental set-up along with the boundary conditions used to describe
the fluid-structure system. The finite element mesh includes: 24 plate-elements
and 560 fluid-elements. The coupled system has a total of 815 active degree of
freedoms. The aluminum base plate is 0.3m long, 0.2m wide and 0.4inm thick
mounted with all its edges in a clamped arrangement in a large aluminum base.
The aluminum base with mounting frame sits on top of a water tank. One side of
the base plate is partially treated with the ACLD/AC/PCLD and the other side is
in contact with water. The material properties and thickness of piezo-electric
material and the viscoelastic layer listed in Table (1). The size of the combined
piezo-electric and viscoelastic patch occupied one-third of the surface area of the
base plate and it is placed in the middle of plate as shown in Figure (3-b). A laser
1264
sensor is used to measure the vibration of the treated plate at node 27 as shown in
Figure (3-b). The sensor signal is sent to a spectrum analyzer to determine the
frequency content and the amplitude of vibration. The signal is also sent via
analog power amplifiers to the piezo-electric layers to actively control the sound
radiation and structural vibration. The radiated sound pressure level into the tank
is monitored by a hydrophone located at 5.0 cm below the plate center. This
position is chosen to measure the plate mode (1, 1) which dominates the sound
radiation. The hydrophone signal is sent also to the spectrum analyzer to
determine its frequency content and the associated sound pressure levels.
Figure (3) - The experimental set-up, (a) schematic drawing, (b) finite element
meshes.
Table (1) - Physical and geometrical properties of the ACLD treatment
Layer
Thickness(m)
Density (Kg/m^)
Modulus(MPa)
viscoelastic
S.OSxlO’"
1104
30**
piezoelectric
28xl0-‘
1780
* Young’s modulus ** Shear modulus
4.2 Experimental Results
Experimental validation of the dynamic finite element model of the
ACLD/plate system in air has been presented by Baz and Ro [15] in detail. Close
agreement was obtained between the theoretical predictions and the experimental
1265
measurements. The d)mamic finite element model is therefore valid to provide
accurate predictions.
For the uncontrolled treated plate/fluid system, considered in this study,
the experimental results indicate that coupling the plate with the fluid loading
results in decreasing the first mode of vibration fi*om 59.475Hz to 10.52Hz. The
coupled finite element model predicts the first mode of vibration to decrease firom
57.91Hz to 10.24Hz. The results obtained indicate close agreement between the
theory and experiments.
Figure (4-a) shows a plot of the normalized experimental vibration
amplitudes for the fluid-loaded plate with the ACLD treatment using different
derivative feedback control gains. According to Figure (4-a), the experimental
results obtained by using the ACLD treatment indicate that amplitude attenuations
of 1 1.36%, 48.25% and 75.69% are obtained, for control gains of 2500, 5000, and
13500, respectively. The reported attenuations are normalized with respect to the
amplitude of vibration of uncontrolled plate, i.e. the plate with PCLD treatment.
Figures (4-b) display the vibration amplitudes of the plate/fluid system with
AC/PCLD treatment at different derivative feedback control gains. The
corresponding experimental attenuations of the vibration amplitude obtained are
4.6%, 20.29%, 54.04% respectively.
Figure (4) - Effect of control gain on normalized amplitude of vibration of the
treated plate, (a) ACLD control and (b) AC/PCLD control.
Figures (5-a) and (5-b) show the associated normalized experimental
sound pressure levels (SPL) using ACLD and AC/PCLD controllers, respectively.
The normalized experimental SPL attenuations obtained using the ACLD
controller are 26.29%, 50.8% and 76.13% compared to 10.02%, 24.52% and
53.49% with the AC/PCLD controller for the considered control gains. Table (2)
1266
lists the maximum control voltages for the ACLD and AC/PCLD controllers for
the different control gains.
It is clear that increasing the control gain has resulted in improving the
attenuations of the plate vibration and the sound radiation into the fluid domain.
It is evident that the ACLD treatment has produced significant vibration and
sound pressure level attenuation as compared to the attenuations developed by the
AC/PCLD or PCLD treatments. It is also worth emphasizing that the ACLD
treatment requires less control energy than the conventional AC/PCLD treatments
to control the sound radiation from the plate.
Figure (5) - Effect of control gain on normalized sound pressure level radiated
from the treated plate, (a) ACLD control and (b) AC/PCLD control.
Table (2) - Maximum control voltage for the ACLD/ AC/plate system
K.
2500
5000
13500
ACLD
0
21.75 V
31.20V
39.60V
AC
50.40V
76.38V
Figure (6) shows the mode shapes of the first four modes of the treated
plate with and without fluid-loading as obtained experimentally using
STARMODAL package. Figure (7) shows the corresponding theoretical
predictions of the first four mode shapes. Close agreement is found between
experimental measurement and theoretical predictions.
Figure (8) presents comparisons between the theoretical and experimental
natural frequencies and the loss factor of a plate treated with the ACLD and
AC/PCLD for different control gains. Close agreement between theory and
1267
experiment is evident. Note also that increasing the control gain has resulted in
increasing the damping ratio for both ACLD and AC/PCLD treatments. The
comparisons emphasize the effectiveness of the ACLD treatment in acquiring the
large damping ratio to attenuate the structural vibration and sound radiation.
Figure (6) - Experimental results of first four mode shapes of treated plate (a)
without fluid loading and (b) with fluid loading.
Figure (7) - Theoretical predictions of first four mode shapes of treated plate (a)
without fluid loading and (b) with fluid loading.
5. SUMMARY
This paper has presented theoretical and experimental comparisons
between the damping characteristics of plates treated with ACLD and
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conventional AC with PCLD treatments. The dynamic characteristics of the
treated plates when subjected to fluid loading is determined for different
derivative control gains. The fundamental issues governing the performance of
this class of smart structures have been introduced and modeled using finite
element method. The accuracy of the developed finite element model has been
validated experimentally. The effectiveness of the ACLD treatment in attenuating
structural vibration of the plates as well as the sound radiated from these plates
into fluid domain has also been clearly demonstrated. The results obtained
indicate that the ACLD treatments have produced significant attenuation of the
structural vibration and sound radiation when compared to PCLD and to AC with
PCLD. Such favorable characteristics are achieved with control voltages that are
much lower than those used with conventional AC systems. The developed
theoretical and experimental techniques present invaluable tools for designing and
predicting the performance of the plates with different damping treatments and
coupled with fluid loading that can be used in many engineering applications.
♦ PCLD ■ ACLD, K:d=l 3500 □ AC, K:d= 13500
• ACLD,Kd=2500 O AC, Kd=2500
A ACLD,Kd=5000 A AC, Kd=5000
Theoretical Natural Frequency (Hz) Theoretical Damping Ratio
Figure (8) - Comparison between theoretical predictions and experimental results,
(a) natural frequency, (b) damping ratio.
ACKNOWLEDGMENTS
This work is funded by The U.S. Army Research Office (Grant number
DAAH-04-93-G-0202). Special thanks are due to Dr. Gary Anderson, the
technical monitor, for his invaluable technical inputs.
1269
REFERENCES
1. Lindholm U. S., Kana, D. D., Chu, W. H. and Abramson, H. N., Elastic
vibration characteristics of cantilever plates in water. Journal of Ship
Research, 1965, 9, 11-22.
2. Fu, Y. and Price, W. G., Interactions between a partially or totally immersed
vibrating cantilever plate and the surrounding fluid. Journal of Sound and
Vibration, 1987, 118(3), 495-513.
3. Ettouney, M. M., Daddazio, R. P. and Dimaggio, F. L., Wet modes of
submerged structures - part litheory. Trans, of ASMS, Journal of Vibration
and Acoustics, 1992, 114(4), 433-439.
4. Kwak, M. K., Hydroelastic vibration of rectangular plates. Trans, of ASME
Journal of Applied mechanics, 1996, 63(1), 110-115.
5. Marcus, M. S., A finite-element method applied to the vibration of
submerged plates. Journal of Ship Research, 1978, 22, 94-99.
6. Chowdury, P. C., Fluid finite elements for added mass calculations.
International Ship Building Progress, 1972, 19, 302-309.
7. Muthuveerappan G., Ganesan, N, and Veluswami, M. A., A note on vibration
of a cantilever plate immersed in water. Journal of Sound and Vibration,
1979, 63(3), 385-391.
8. Rao, S. N. and Ganesan, N., Vibration of plates immersed in hot fluids.
Computers and structures, 1985, 21(4), 111-1%! .
9. Everstine G. C., Prediction of low frequency vibrational frequencies of
submerged structures. Trans, of ASME, Journal of Vibration and Acoustics,
1991, 113(2), 187-191.
10. Jones, I. W. and Salerno, V. L., The vibration of an internally damped
sandwich plate radiating into a fluid medium. Trans, of ASME, Journal of
Engineering for Industry, 1965, 379-384.
11. Sandman B. E., Motion of a three-layered elastic-viscoelastic plate under
fluid loading. J. of Acoustical Society of America, 1975, 57(5), 1097-1107.
12. Dubbelday, P. S., Constrained-layer damping analysis for flexural waves in
infinite fluid-loaded plates. Journal of Acoustical Society of America, 1991,
(3), 1475-1487.
13. Gu, Y. and Fuller, C. R., Active control of sound radiation from a fluid-
loaded rectangular uniform plate. Journal of Acoustical Society of America,
1993, 93(1), 337-345.
14. Shen, I. Y., Bending vibration control of composite plate structures through
intelligent constrained layer treatments. Proc. of Smart Structures and
Materials Conference on Passive Damping ed. C. Johnson, 1994, Vol. 2193,
115-122, Orlando, FL.
1270
15. Baz, A. and Ro, J., Vibration control of plates with active constrained layer
damping. Journal of Smart Materials and Structures, 1996, 5, 272-280.
16. Craggs, A., The transient response of a coupled plate-acoustic system using
plate and acoustic finite elements. Journal of Sound and Vibration, 1971,
15(4), 509-528.
APPENDIX
1. Stiffness Matrix of the Treated Plate Element
The stiffoess matrix [Kp]; of the ith element of the plate/ACLD system is
given by Baz and Ro [15]:
(A-1)
where [K,l and [KJi denote the in-plane, shear and bending stifj&iesses of
the ith element. These stiffoess matrices are given by:
j = layer 1,2, and 3 (A-2)
and [K.,1 = I jjB,]"[Dj,][B,]dxdy j = layer 1, 2 and3 (A-4)
with G2 denoting the shear modulus of the viscoelastic layer and the matrices [BJ,
b1 = :^
\({N2}-{N4)/d + {N,}
(n,1 +fNj
’ [Bb] =
2{N5},„_
{Naj
[B.] =
{n.
[®^p] =
(N.},. + {N3},.+h{Ns},„
{n.},, + + {N2}^ +{n.) „ +h{N4^,
1271
- 1
0
_ i
1 -
0
Ei
1 n
and fj, 1 EA
1 n
1-vJ
Vj 1 u
Vj 1 u
I_
' « '“Vi
0
0
i'
_o 0 V.
. j=l,2and3 (A-5)
where h = (hi-h3)/2 and d = (h2+hi/2+D) with D denoting the distance from the
mid-piane of the plate to the interface with the viscoelastic layer. Also, Ij
represent the area moment of inertia of the jth layer.
2. Mass Matrix of the Treated Plate Element
The mass matrix [Mp]; of the ith element of the plate/ACLD system is
given by:
(A-6)
where [Mjp]j and [M^Jj denote the mass matrices due to extension and bending of
the ith element. These matrices are given by
[Mi,].=p,h,££({N,}^{N,}+{N3}''{N,})dxdy + p,h,[ [({N3)'"{N3} + {NX{N4))dxdy
+ ip2h2 £ { +{N,}^{N,})dxdy
and [ ], = (p,h, + P3h3 + P3h3) £ £ [ N3 f [ N3 ]dxdy (A-7)
where {NJ = {N,}+{N3}+h{Ns},3 and {N,} = {NJ+{N4}+h{N5},3
3. Control Forces and Moments Generated by the Piezo-electric Layer
3.1 The in-plane piezo-electric forces
The work done by the in-plane piezo-electric forces {Fp}i of the ith
element is given by:
i{5'}-{Fp}rhi££%d>'dy (A-8)
where j=l for ACLD control or j=3 for AC control. Also, Ojp and Sjp are the in¬
plane stresses and strains induced in the piezo-electric layers. Equation (A-8)
reduces to:
1272
='"•1 IKFK]
for k=l, 4
3.2 The piezo-electxic moments
The work done by the piezo-electric moments {Mp}i due to the bending of
the piezo-electric layer of the ith element is given by:
5' .MM
= hj U,e,Ady
(A-10)
Where Gjb and are the bending stresses and strains induced in the piezo-electric
layers. Equation (A-10) reduces to:
Mpxk
x.'
Mpyk
d32
Mp,yk
_ 0
fork=l,..,4
(A-11)
4. Stiffness and Mass Matrices of the Fluid Element
The stiffness matrix [KJj and mass matrix [MJ^ of the ith element of the
fluid system are given by:
[B,rNdv
(A-12)
(A-13)
[N,]>,]dv (A-13)
where [b^ ] = [[N^],, [N^] ^ [N^] and c is the sound speed.
5. Coupling matrix of the Treated Plate/Fluid System
The coupling matrix [Q] of the interface element of the structure/fluid
system can be presented by:
[q]= f [ {N5}''[Nf]dxdy (A-14)
Ja Jb
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1274
ANALYTICAL METHODS II
DYNAMIC RESPONSE OF SINGLE-LINK FLEXIBLE
MANIPULATORS
E. Manoach^ G. de Paz^ K. Kostadinov^ and F. Montoya^
^ Bulgarian Academy of Sciences, Institute of Mechanics
Acad. G. Bonchev St. Bl. 4; 1113 Sofia, Bulgaria
^ Universidad de Valladolid, E.T.S.I.L Dpto. IMEIM.
C/Paseo del Cauce, s/n 47011- Valladolid, Spain
1. INTRODUCTION
The flexible-link manipulators have many advantages over the traditional stiff
ones. The requirements for light-weight and energy efficient robotic arms
could be naturally satisfied by using flexible manipulators. On the other hand
the application of the robotic arm in such activities as positioning in electronic
microscopes and disc-drivers, hammering a nail into a board or playing tennis,
also forces the modeling and control of the dynamic behavior of flexible link
manipulators.
In most cases the elastic vibrations which arise during the motion must be
avoided when positioning the end point of a robotic arm. These are a part of
the reasons that cause a great increase of the publications in this topic in recent
years.
In most papers the flexible robotic arms are modeled as thin linear elastic
beams. In [1-3] (and many others) the Bemouli-Euiler beam theory, combined
with finite-element technique for discretization with respect to the space
variables is used for modeling and control of single-link flexible manipulators.
The same beam theory, combined with mode superposition technique is used
in [4]. Geometrically nonlinear beam theories are used in [5,6] for the
modeling of a single-link and multi-link flexible robotic manipulators,
correspondingly.
Taking into account the fact that robotic arms are usually not very thin and that
the transverse shear could play an important role for dynamically loaded
structures [7] the application of the Bernouli-Euiler beam theory could lead to
a discrepancy between the robotic arm behavior and that one described by the
model.
1275
The aim of this work is to model the dynamic behavior of a single link flexible
robotic arm employing the Timoshenko beam theory, which considers the
transverse shear and rotary inertia. The arm is subjected to a dynamic loading.
As in [3], the viscous friction is included into the model and slip-stick
boundary conditions of the rotating hub are introduced. Besides that, the
possibility of the rise of a contact interaction between the robotic arm and the
stop (limiting support) is included into the model. The beam stress state is
checked for plastic yielding during the whole process of deformation and the
plastic strains (if they arise) are taken into account in the model. The
numerical results are provided in order to clarify the influence of the different
parameters of the model on the response of the robotic arm.
2. BASIC EQUATIONS
2.1. Formulation of the problem
The robotic arm - flexible beam is attached to a rotor that has friction and
inertia. The beam is considered to be clamped to a rotating hub and its motion
consists of two components: “rigid-body” component and a component
describing the elastic deflection of the beam (see Figure 1). The motion of the
flexible beam is accomplished in the horizontal plane and gravity is assumed
to be negligible.
Figure 1. Model of one-link flexible manipulator.
1276
Tip of the beam (with attached tip-mass) is subjected to an impulse loading.
Stick-slip boundary conditions due to Coulomb friction of the hub are
introduced when describing the motion of the beam. In other words, if the
bending moment, about the hub axis, due to the impact is lower than the static
friction torque threshold then the hub is considered clamped and the beam
elastic motion is considered only. When the bending moment exceeds friction
torque threshold this boundary condition is removed, allowing rotation of the
hub and the arm. When the hub speed and kinetic energy of the beam become
again beneath the torque threshold, the hub clamps again.
The possibility of the rise of a contact interaction between the robotic and the
stop is envisaged. In this case, if the hub angle exceeds the limit value, the
robotic arm clamps and a part of the beam goes in a contact with the stop,
which is modeled as an elastic foundation of a Vinkler type.
In view of the fact that the impact loading and contact interactions are
included in the investigations, it is expedient to be considered the rise of
plastic strains in the beam.
2.2. Deriving the equations of motion.
The total kinetic and potential energy of the rotating hub with the attached
beam (described by the Timoshenko beam theory) and a lumped mass at it’s
tip can be expressed as follow:
Ek =^|pA[u(x,t)] dx + |j„[e(t)]^+tMT[u(l,t)f jEl[(()(x,t)] dx
EIl^l +kGA’
dx.
V dx
dx
(1 a,b)
In these equation u(x,t) is the total displacement
u(x,t) = w(x,t)+x0(t) (2)
and w is the transverse displacement of the beam, (p is the angular rotation,
9 is the hub angle, E is the Young modulus, G is the shear modulus. A: is a
shear correction factor, p is the material density, Jh is the inertia moment of
the hub, A=b*h is the beam cross-section area, h is the thickness, b is the
width, I=bh^/12 , 1 is the length of the beam, Mt is the tip mass, t is the time.
1277
Denoting the work of external forces (applied actuating torque T(t) and the
beam loading p(x,t))by
W=T(t)e(t)+ \p{x,t)w{,x,t)dx
0
the Hamilton's principle can be applied:
j5(E,-Ep)dt+ j5Wdt = 0
(3)
(4)
Substituting eqns (l)-(3) into eqn (4) after integrating and including damping
of the beam material, the viscous friction of the hub and the reaction force of
the elastic foundation the following equations of motion can be obtained:
. r . X. i2x de , ,5^w(l,t) ,
(J + Jh + M-j-1 ^ j ^2 ^ ^ ^ ^2
i
JpAx
a^w(x,t)
dt^
dx = T(t)
EI^-^ + kGA
dx^
dw
dx
dw
.d^cp
- 9 “ <^2 - — 2 ~ 0<x</ , t>0 (5a-c)
dt
dt
kGA
+ x-
a^0
'St"
-R(x,t) = p(x,t)
The boundary conditions are:
w(0,t)=(p(0,t)=0
dcpgt) Q
dx
kGA
dw(l,t)
dx
d^w(lt)
dt^
(6a-c)
and the initial conditions are:
wfx, 0) =w^(x), w(x,0) = w° (x) , p(x, 0)=(f^(x), (p(x,0) = 9° (x)
e(t) = e(t) = o, t<t,,p
or 9(0 =
when |9(t)| and kinetic energy of the beam falls simultaneously under the
Coulomb friction thresholds.
In eqns (5) the viscous friction coefficient of the hub is denoted by di, d2 and
d^ are damping coefficients of the beam material, J is the moment of inertia of
the beam about the motor axis and R(x,t) is the reaction force of the stop
disposed from Xj to X2 (xj<X2<l) and modeled as an elastic foundation with
Vinkier constant r:
(7a-d)
(7 e,f)
(7e)’
1278
R(x,t) =
r[w(x, t) - (x)] for x j < x < x^ ; |0l > '
0 for 0<x<x,,X2 <x</; iBi <
The time when the bending moment about the hub axis exceeds the friction
torque threshold is denoted by tsUp^ When t>tsiip this condition is removed
(allowing rotation of the hub and the beam) until the moment when the beam
clamps again.
2.3. Elastic-plastic relationships
The beam stress-strain state is usually expressed in terms of generalized
stresses and strains which are function of x coordinate only. As a unique
yield criterion in terms of moments and the transverse shear force does not
exist according to Drucker [8], the beam cross-section is divided into layers
and for each of them the stress state has to be checked for yielding.
The relation between the stress vector S = and the strain vector
s = |-z ^ I ’ generally presented as
S=[D]8,
where in the case of an elastic material [D] = [D^] =
is the elastic
matrix and f(z) is a function describing the distribution of the shear strains
along the thickness.
On the basis of the von Mises yield criterion, the yield surface is expressed
After yielding during infinitesimal increment of the stresses, the changes of
strains are assumed to be divisible into elastic and plastic parts
As = As^ + As^
where
As
1279
By using eqns (10), (11) and the associated flow rule [9], following Yamada
and others [10], the following explicit relationship between the increments of
stresses and strains is obtained
. AS = [d‘'’]A8
where [-0'^] is the elastic-plastic matrix:
HY\d^]
LjaLiVi
r^r
1 J 1 J L j\asj
1 -
C/2
1—
108]
In this equation H is a function of the hardening parameter. For ideal plasticity
H is equal to zero, while for a wholly elastic material H -> co .
3. Use of mode superposition.
3.1. Rearrangement of the equation of motion.
Let the total time interval T on which the dynamic behavior of the structures is
investigated, be divided into sequence of time increments y+J.
In the numerical calculations the following dimensionless variables are used:
x = xl U w = w//, t =tl / c c = / p
and then omitting the bars, and after some algebra, the governing equations
can be written in the following form:
d'e de
- T- + C, -
d t^ ' d t
--cJ w(l,t)- f(p(x,t)dx =C3 T(t)+ fxp(x,t)dx
tv 0 /Vo /
09 0^9 -I c/w I
(15 a-c)
0^W 01V 0^iv 09
dt^ ^ dt 1 0x ^ dx
= -p~G{-G^
where a=12P/h^ , p~kG/E, p=pI/(EA), c,= d^l/(cJ}{), Arp/(JHp) C3=
P/(c^J}^ ), C4~d49/EI, C3=djl/EA, =5/6. The nonlinear force due to the
reaction of the foundation is denoted by G^ = R(x,t).l/EA and Gf and G2 are
the components of the so-called non-linear force vector Gp( .G^} which is
due to the inelastic strains. It has the presentation (see [7] ):
1280
3.2 Mode superposition method
The l.h.s. of eqns (15 b,c) is a linear form and therefore the mode
superposition method can be used for its solution. As the eigen frequencies
and the normal modes of vibrations of an elastic beam do not correspond to the
real nonlinear system, these modes are called "pseudo-normal" modes.
Thus, the generalized displacements vector v = |a”^ (p, w| is expanded as a
sum of the product of the vectors of pseudo-normal modes v^ and the time
dependent functions q„(t) as
(17)
n
The analytically obtained eigen functions of the elastic Timoshenko beam are
chosen as basis functions (see APPENDIX). When the tip mass is attached to
the beam the eigen functions of the system used in the mode superposition
method are preliminary orthogonalized by standard orthogonalization
procedure as it is mentioned in the APPENDIX.
Substituting eqn (17) into (15 b,c), multiplying by v^(x), integrating the
product over the beam length, invoking the orthogonality condition (see (A9) )
and assuming a proportional damping for the beam material
|(c4(pf, +c^wl)rdr the following system of ordinary differential
equations (ODE) for 0(t) and qn(t) is obtained:
^(f) + c, (9(0 = Cj 2] [«'» (1) - ]?»
^ »
+ C3(r(0 + P(0)
(18a, b)
9„(o+2f„ffl„9„(o+®,k,(o=-‘y„ -'f„&(o+&!’
In this equations
1
11 V f
Y„ = J(p„ 6„ = = jxw,Xx)ck, P(t)= \xp{x,t) dx ,
1 1
g;(t)= |G'’(x,t)v„(x)dx, g',(t)= jG'(x.t)w„(x)dx, co„ are the eigen
0 0
frequencies of the elastic clamped beam and are the modal damping
parameters.
The initial conditions defined by eqns (7 a-d) are transformed also in terms of
q„(0), and ?„(0)
9„(0) = 9„“, 4„(0) = 4°,
q°„ = J(w°w„ t-a'VVjdx, (j“„ = j(w"w„ +a-'(p°(p„)dx, (19a-d)
0 0
The obtained system of nonlinear ordinary differential equations is a stiff one
and it is solved numerically by the backward differential formula method, also
called the Gear's method [11].
The rise of plastic strains is taking into account by using an iterative procedure
based on the "initial stresses" numerical approach [7].
4. RESULTS AND DISCUSSION
Numerical results were performed for the robotic-arm with the same material
and geometrical characteristic as these given in [3] in order to make some
comparisons. Model parameters are: E=6.5xl0 Pa ,v=0.2, Cp=2.6xl0 Pa,
1=0.7652, b=0.00642 m, h=0.016 m, p=2590 kgW , Mt=0,153 kg, Jh=0.285
kgm^ (Jh is not defined in [3].)
The aim of the computations is to show and clarify the influence of the elastic
or elastic-plastic deformation on the motion of the robotic-arm, to demonstrate
the effect of the hub friction, slip-stick boundary conditions and the contact
interaction between the beam and the stop.
1282
Only impact loading on the beam is considered in this work, i.e. dynamic load
p(x,t) and applied torque T(t) are equal to zero. The impact loading is
expressed as an initial velocity applied to the tip of the beam 0.95 < x < 1 .
Nine modes are used in expansion (15) but the results obtained with number of
modes greater than nine are practically indistinguishable from these shown
here.
For all calculations the material damping is equal to 8% of the critical
damping.
The results for the rotation of the hub of the flexible manipulator with an
attached mass at its tip -1 and without an attached mass -2 are shown in Figure
2. The hub friction is not considered. The beam is subjected to an impact
loading with initial velocity =-1.95 m/s. As can be expected, the hub angle
increases much faster in the case of the beam with an attached mass. The
corresponding beam deflections are presented in Figure 3. The results obtained
are very close to these obtained in [3] (Fig. 7 and 8 in [3]). The frequencies of
forced vibrations obtained in [3], however, correspond to the beam without an
attached mass.
0, rad
-4.00 I - ^ ^ ^ ^ - 1 - ^ ^ i ^
0,00 l.OO 2.00 3.00 4.00 5.00
t, sec
Figure 2. Hub response without
viscous friction. 1 - beam with an
attached mass; 2- without an attached
mass
w, m
Figure 3. Deflection of the tip of the
beam without viscous friction of the
hub. 1 - beam with an attached mass;
2 - without an attached mass
1283
The influence of the hub friction on the flexible manipulator response can be
seen in Figure 4. The viscous friction is set di=0.1 Nms, the static Coulomb
friction threshold is equal to 0.06 Nm and three cases of the hub slip-stick
threshold are tested: |e| stick = 0-005 rad.s'^ - 1, |0| stick = 0.0085 rad.s'^ ,
stick = 0.01 rad.s'^ For this initial velocity (w° =-1.95m/s) the hub slips
very fast from the initial clamped state and the beam begins to rotate. As can
be expected, the consideration of the viscous friction of the hub leads to a
decrease of the angle of rotation of the beam and changes the linear variation
of 0 with time. The results show also that the value of the hub speed
threshold |0| stick exercises an essential influence on the motion of the
rotating system.
When |e| =0.005 rad.s' the
beam sticks at t = 2.602 s after that
the hub periodically slips and sticks
which also leads to damping of the
motion. When joj stick “ 0.0085
rad.s'^ the start of sticking occurs at
t = 1.7207 s and after t=2.417 s the
hub clamps with short interruptions
till t=3.4 s when due to the elastic
vibrations it snaps in the direction
opposite to w°» clamps again at
3.679s, slip at 4.5s, and finally
clamps at t=4.5s. When joj stick
=0.0085 rad.s"' the sticking begins
at t= 1.0525s and very fast (at t«2 s)
the beam clamps with 0=-O.587 rad.
In order to observe the occurrence of the plastic deformation the beam was
subjected to impulse loading having larger values of initial velocities. In
addition, the contact interaction between the beam and the stop was
considered.
The beam-tip deflection in the presence of a contact with the stop disposed at
X €[0.16, 0.263] and initial velocity w° =-15.95 m/s is shown in Fig. 5.
0,rad
di=0.1Nms. 1- |0| stick”0-005rad.s"',
2 - |0| stick = 0-0085 rad s"' ;
3- |0| stick =0-0 is-'
1284
In order to reduce the computational time the limit value of 0 was chosen
e^j^p=0.0025 rad. When this value was reached the problem was automatically
resolved with new initial conditions 0=0, w°=w(Xstop,tstop), etc.
As can be seen, the presence of the contact interaction during the process of
motion of the beam due to the elastic support for x e[0.16, 0.263] leads to a
decrease of the amplitudes of vibrations in the direction of the stop in
comparison with the amplitudes in the opposite direction. The variation of the
beam displacements along the beam length for the first 0.8 s of motion is
shown in Fig. 6. It must be noted that in this case of loading a plastic yielding
occurs. It is assumed that beam material is characterized by an isotropic linear
strain hardening and H=0.5. The plastic yielding occurs simultaneously with
the contact interaction at t=0.01366 s at the clamped end of the beam. At
t=0.0186 s the plastic zone spreads to x = 0.158 and at t=0.08767 if covers the
length to X =0.3 1 . The last points that yields are x =0.55, 0.61 at t=l . 1 19 s.
Seven layers along the beam
thickness, symmetrically disposed
about the beam axis was checked for
yielding (N2=7) but the plastic zone
has reached the second and 6th
layers only at the clamped end of the
beam (x =0). In all other point along
beam length the plastic yielding
occurs only at the upper and lower
surface of the beam. The plastic
strains are small and the response of
the beam is not very different from
the wholly elastic response.
Nevertheless, the appearance of such
kind of plastic deformations in the
structures used for the precise
operations must be taken into
account in the manipulator self
calibration procedure.
w,m
0.40
0.30
0.20
0,10
0.00
-O.IO
-0.20
-0.30
0.00
1.00
2.00
3.00
4.00 5.00
t, sec.
Figure 5. Deflection of the tip of the
beam with time in the case of a
contact with the stop. r=5.5xl0^ Pa
5. CONCLUSIONS
In this work a model describing dynamic behavior of a deformable beam
attached to a rotating hub that has friction and inertia is developed. The
Timoshenko beam theory is used to model the elastic deformation of the beam.
1285
Figure 6, Variation of the elastic-plastic beam displacement along the beam
length with time in the case of a contact with the stop.
The slip-stick boundary conditions are also incorporated into the model
The possibility of the rise of undesired plastic deformations in the case of a
high velocity impact on the clamped robotic arm, or in the case of a contact
with limiting support (stop) is included into the model.
The analytically obtained eigen functions of the elastic Timoshenko beam
vibrations are used to transform the partial differential equations into a set of
ODE by using the mode superposition method. This approach minimizes the
number of ODE which have to be solved in comparison with another
numerical discretization techniques (finite elements or finite difference
methods).
The results obtained show the essential influence of elasticity on the robotic-
arm motion.
The model will be used to synthesise a control of one link flexible
manipulators and for a self calibration procedure when plastic deformation
would occur.
Acknowledgments
The authors gratefully acknowledge the financial support from EC Copernicus
Program under the Project ROQUAL CIPA CT 94 0109.
The first author wishes to thank the National Research Fund for the partial
financial support on this study through Contract MM-5 17/95.
1286
REFERENCES
1. Bayo, E. A finite-element approach to control the end-point motion of a
single -link flexible Robert. J. Robotic System, 1987, 4„ 63-75
2. Bayo, E. and Moulin, H., An efficient computation of the inverse dynamics
of flexible manipulators in the time domain. IEEE Proc Int. Conf. on
Robotics and Automations, 1989, 710-15.
3. Chapnik, B.V., Heppler, G.R., and Aplevich, J.D. Modeling impact on a
one-link flexible robotic arm. IEEE Transaction on Robotics and
Automation, 1991,7,479-88.
4. Liu, L. and Hac, A., Optimal control of a single link flexible manipulator.
Advances in Robotics, Mechatronics, and Haptic Interfaces, 1993, DCS-
49, 303-13.
5. Wen, J.T., Repko, M. and Buche, R., Modeling and control of a rotating
flexible beam on a translatable base. Dynamics of Flexible Multibody
Systems: Theory and Experiment, 1992, DCS-37, 39-45.
6. Sharan, A.M. and Karla, P., Dynamic Response of robotic manipulators
using modal analysis. Meek Mach. Theory, 1994, 29, 1233-49.
7. Manoach, E. and Karagiozova, D. Dynamic response of thick elastic-plastic
beams. International Journal of Mechanical Sciences, 1993, 35, 909-19
8. Drucker, D.C. Effect of shear on plastic bending of beams. J. of Applied
Mechanics, 1956, 23, 515-21
9. Hill, R. Mathematical Theory of Plasticity, 1950, Oxford University Press,
London.
10. Yamada, Y., Yoshimura, N. and Sakurai T., Plastic stress-strain matrix and
its application for the solution of elastic-plastic problems by the finite
elements, Int. J. of Mechanical Sciences , 1968, 10, 343-54
11. Gear, C.W., Numerical initial value problem in ordinary differential
equations, 1971, Prentice-Hall, Englewood Cliffs, NJ.
12. Abramovich, H. Elishakoff, 1. Influence of shear deformation and rotary
inertia on vibration frequencies via Love’s equations. J. Sound Vibr., 1990,
137, 516-22.
1287
APPENDIX: NORMAL MODES OF FREE VIBRATIONS OF A
CLAMPED TIMOSHENKO BEAM WITH AN ATTACHED MASS,
Equations (5 b,c) can be decoupled, transforming them into two fourth order
equations [12] as regards cp and w.
Solving this equations (with p=0 and R=0) and using the boundary conditions
(6) (with 0=0) , the equations of the frequencies and forms of vibrations of the
beam are obtained.
Introducing following denotations
1/2 (A1 a-e)
^2„={®,^(l + P) + [®:a + P)^+4(B,^(a-pco,^,)]''^} /2,
/l„ =(4+CoJP)/5,„, /2„ /3„ =(4-®SP)''-S!».
the frequencies of free vibrations are determined as roots of equation:
a) In the case >0 i.e. co < a / p the frequencies equation is:
^11^22 ”^12^21 “ ^ > (-^2)
where
^11 ~ /2n‘^2n ^12 ~ ■^/2«‘^2« ®^^(‘^2«)
h\ = -f\nM.Sxn ) “ /2„ sin(52„ ) + s^„sh(si, ) + 52„ sin(52„ ) + , . .
(A3 a-d)
M‘^lnCh(^lJ + ‘^2« cos(,y2J]
*22 = /2j=OS(^2«)-cll('SlJ] + 4^S,„ch(i,„)-52„ COS(52„) +
J\n
+ sl„ sin(i2„)]
Jin
and the modes of vibrations are:
w„W = -S„
1
' b
^(ch(i,„x) - cos(j2„x)) + sin(i2„x) - ^sh(i|„x)
J\»
L^i
(A4 a,b)
(/,„sh(,s,„x) + sin(,y2«^)) + fi,, icos{s^„x) - ch(s,„x)
b) In the case 5^,, <0 i.e. > a / P the eigen frequency equation (A2) has
the following presentation:
1288
b\\ = + /2„i2„ C0S(J2„)
bn = ?i„/2nSin(?i„) - /2„i2n sin(j2„)
621 = /3„sin(ii„) - /2„ sin(52„) -?i„sin(Ji„) + S2„ sinC^jJ -
>.[J,^„COS( j,„ ) - sl„ C0S(i2„)]
*22 = /2„[cOS(S2„) - COS(J,„)] - ^ S,„COS(?,„) + S2„ COS{s-^„) -
J\n
- sl„ sin(52„)]
7 In
sh = -5?„
and the of vibrations are:
w„(x) =
f b
sin(j2„2c) - ^sin(?,„x) - ^(cosCij,,*) - cos(j,„x))
/i„ *11
'P„W = -S»
(A5 a-e)
(A6a-b)
/j,, (cos(i2„x) - cos(^|„x)) - ^(/3„sin(i,„x) - f^„ sin(jj„x))
0,,
When a mass is not attached at the beam tip the following orthogonality
condition is fulfilled:
1
p fO, n^ m;
(a _
J [1, n - m.
(A7)
and when an attached mass is considered the modes are orthogonalized by
standard orthogonalization procedure.
The constants are obtained from condition (A7).
1289
1290
Wave Reflection and Transmission
in an Axially Strained, Rotating Timoshenko Shaft
B. Kang ‘ and C. A. Tan ^
Department of Mechanical Engineering
Wayne State University
Detroit, Michigan 48202, U. S. A.
Abstract
In this paper, the wave reflection and transmission characteristics of an axially strained,
rotating Timoshenko shaft under general support and boundary conditions, and with geometric
discontinuities are examined. The static axial deformation due to an axial force is also included
in the model. The reflection and transmission matrices for incident waves upon these point
supports and discontinuities are derived. These matrices are combined, with the aid of the
transfer matrix method, to provide a concise and systematic approach for the free vibration
analysis of multi-span rotating shafts with general boundary conditions. Results on the wave
reflection and transmission coefficients are presented for both the Timoshenko and the simple
Euler-Bemoulli models to investigate the effects of the axial strain, shaft rotation speed, shear
and rotary inertia.
‘ Graduate Research Assistant. Tel: +1-313-577-6823, Fax: +1-313-577-8789. E-mail: kang@feedback.eng.wayne.edu
’ Associate professor (Corresponding Author),Tel: +1-313-577-3888, Fax:+1-313.577-8789. E-mail: tan@tan.eng.wayne.edu
Submitted to: Sixth International Conference on Recent Advances in Structural Dynamics, Institute of Sound and Vibration,
Southampton, England, July, 1997
1291
A.
NOMENCLATURE
Area of shaft cross section [m^]
do
Diameter of shaft cross section [m]
C
Generalized coordinate of an incident wave [m]
Cdt {Ct)
Translational damping coefficient [N-sec/m]
Cdr (Cr)
Rotational damping coefficient [N-m-sec/rad]
Co
Bar velocity [m/sec]
Cj
Shear velocity [m/sec]
D
Generalized coordinate of a transmitted wave [m]
E,G
Young’s and shear modulus [NW], respectively
I
Lateral moment of inertia of shaft [m'^]
Jm (Jm)
mass moment of inertia of a rotor mass [kg-m"^]
K
Timoshenko shear coefficient
Knikr)
Rotational spring [N/rad]
Kt (kd
Translational spring [N/m]
t
Length of shaft [m]
M{m)
Mass of rotor [kg]
P
Axial force [N]
rij , ti]
Reflection and transmission coefficients, respectively, i - 1 positive traveling
U{u)
wave; i = 2 negative traveling wave; jr = 1 propagating wave for Cases 11 and
/V; 7 = 2 attenuating wave for Cases II and IV. Both 7 =1 , 2 for propagating
wave for Case I
Transverse displacement [m]
X-Y-Zix-y-z)
Reference frame coordinates [m]
a
{K-G)IE
Rotation parameter, see Eqn. (Id)
£
Pl{E-A), axial strain
e'
Non-dimensional axial load parameter, see Eqn. (13b)
T, f (r, y)
Wavenumber [m‘‘]
nAi)
See Eqns. (20a, b), (22a, b) and (24a, b)
p
Mass density of shaft [kgW]
G
Diameter ratio between two shaft elements
57, (ft))
System natural frequency for Timoshenko model [rad/sec]
a
System natural frequency for Euler-Bernoulli model [rad/sec]
Q.
Rotation speed of shaft [rad/sec]
W (¥)
Bending angle of the shaft cross-section [rad]
subscript L, r
The left and right side of a discontinuity, respectively.
superscript -,+
Negative and positive traveling waves, respectively, when used in C and D.
Note: Symbols in
Otherwise denotes quantities on the left and right side of a discontinuity,
respectively
parenthesis are the corresponding non-dimensional parameters.
1292
1. INTRODUCTION
The vibrations of elastic structures such as strings, beams, and plates can be described in
terms of waves propagating and attenuating in waveguides. Although the subject of wave
motions has been considered much more extensively in the field of acoustics in fluids and solids
than mechanical vibrations of elastic structures, wave analysis techniques have been employed to
reveal important, physical characteristics associated with vibrations of structures. One advantage
of the wave technique is its compact and systematic approach to analyze complex structures such
as trusses, aircraft panels with periodic supports, and beams on multiple supports [1]. Previous
works based on wave propagation techniques have been well documented in several books [2-4],
Recently, Mead [5] applied the phase-closure principle to determine the natural frequencies of
Euler-Bemoulli beam models. A systematic approach including both the propagating and near-
field waves was employed to study the free vibrations of Euler-Bemoulli beams [6].
High speed rotating shafts are commonly employed in precision manufacturing and power
transmission. Despite the usefulness of the wave propagation method in structural vibrations,
applications of this technique to study the dynamics and vibrations of a flexible shaft rotating
about its longitudinal axis have seldom been considered. The purpose of this paper is to examine
the wave reflection and transmission [6] in an axially strained, rotating Timoshenko shaft under
various support and boundary conditions. The effect of the axial load is included by considering
the axial static deformations in the equations of motion. This paper is a sequel to another paper
in which the authors discuss the basic wave motions in the infinitely long shaft model [7].
Although there have been numerous studies on the dynamics and vibration of rotating shafts,
none has examined the effects of axial strains (which cannot be neglected in many applications)
on the vibration characteristics of a Timoshenko shaft under multiple supports. Modal analysis
technique has been applied to study the vibration of a rotating Timoshenko shaft with general
boundary conditions [8, 9], and subject to a moving load [10]. Recently, the distributed transfer
function method was applied to a rotating shaft system with multiple, geometric discontinuities
[11], The wave propagation in a rotating Timoshenko shaft was considered in Ref. [12]. Other
major works on the dynamics of rotating shafts have been well documented in Refs. [13-15].
This manuscript is organized as follows. Governing equations of motion [16] and basic wave
solutions for the Timoshenko shaft are outlined in Section 2. Each wave solution consists of four
wave components: positive and negative, propagating and attenuating waves. In Section 3, the
wave reflection and transmission matrices are derived for the shaft under various point supports
and boundary conditions. The supports may include translational and rotational springs and
dampers, and rotor mass. Results are presented for both the Timoshenko and the simple Euler-
Bemoulli models to assess the effects of axial strain, shaft rotation, shear and rotary inertia. The
wave propagation across a shaft with geometric discontinuities such as a change in the cross-
section is examined in Section 4, and the wave reflection at a boundary with arbitrary support
conditions is considered in Section 5.
With the wave reflection and transmission matrices as the main analytical tools, it is shown in
Section 6 how to apply the current results together with the transfer matrix method to analyze the
1293
free vibration of a rotating, multi-span Timoshenko shaft system in a systematic manner. The
proposed approach is then demonstrated by considering the free vibration of a two-span beam
with an intermediate support.
2. FOKMULATION AND WAVE SOLUTIONS
Consider a rotating shaft subjected to axial loads and with multiple intermediate supports and
arbitrary boundary conditions, as shown in Fig. 1. Including the effects of rotary inertia, shear
deformations, and axial deformations due to the axial loads, the uncoupled equations of motion
governing the transverse displacement u and the slope \}/ due to bending can be derived in the
following non-dimensional form
.. . d'^u „ d^u
£
+16a(l + £)(l-i-e - )^r^ = 0 ,
a dr
U lA £ . d~U
■2ri3^ + a— -16£{1 + £--)^^
a dz'^
-l-16a(l-l-£)(l + £-— = 0 ,
a dr
u = — z = — r = — T = ^ ■
a.' a.' T. ^ \ KG
Figure 1. A rotating Timoshenko shaft model subject to axial loads and with general boundary conditions.
1294
(Id)
a
KG
E ’
E
Note that u and y/are the measurements in the complex plane, that is u=ux+iuy and \i/=\}fx+iYr
E denotes the Young’s modulus, p the mass density, As the area of the cross section, ao the
diameter of shaft, K the Timoshenko shear coefficient, G the shear modulus and Q the constant
angular velocity of the shaft. Details of deriving these equations of motion are found in Ref.
[16].
Assuming and substituting the following wave solutions into Eqns. (la) and (lb)
= (2a)
= (2b)
and defining the non-dimensionalized wavenumber 7 and system natural frequency w gives the
frequency equation, Eqn. (3a); see Ref. [7],
y = r^o .
_ CO a,
KG .
(j) = - a. = - is known as the shear velocity) .
where,
7“* -A7" 4-5 = 0 ,
A = (1 -i- a)a) ^ - 2j3 cu “ 16£ (1 + e - ) ,
5 = a ot)^ - 2j3 u) - 16a (1 4- £)(1 4- e - •^)
The four roots of Eqn. (3a) are
7 = ±-^[a ± -x/a^ -45 .
(2c)
(2d)
(3a)
(3b)
(3c)
(4)
In general, 7 is complex. Let (o be real. It can be shown that, with a > 0 and £ the axial strain
of the elastic solid, the discriminant A^ -45 is positive semi-definite for most engineering
applications. Hence, it is possible to classify the wave solutions into four distinct cases. Note
that one may study the wave propagation by considering only a single general form of the wave
solution. However, the classification procedure identifies the coupled modes of vibration of the
1295
Timoshenko shaft model and provides a better understanding on how each wave solution governs
the wave motions [7]. Based on the algebraic relationships between A and B, the four valid wave
solutions are obtained as follows.
Case / ( A > 0 and B > 0 );
«(z, t) = + C- (5a)
V(z, t) = (CJ.e-'*’" + )e® (5b)
Case !I{A>0 andB<0):
«(z, t) = (Ce-"''' + + C:^e-^'-‘ + (6a)
yz(z,t) = (c;,^-'^" - c;,/" + + 0;^/=- )£® (6b)
CaseIIIiA<0mdB>0):
uiz,t) = (C>-''' + + C:,«")«'“ (7a)
Vr(z,0 = (C;,a-" + C;,ef>' + + C;je'=')«® (7b)
Coje/V(A<0 andB<0):
k(zA) = (C,>-^'^ +C,>’‘'- + C>-'''=‘ + C,V'’'*')«'® (8a)
V/(z,«) = (C;,e-f- + (8b)
where,
f, =^(|A| + V-5’-4|B|f, r. (9^. W
r, = + 4|B| + |A|)^ , r, + 4|B| - |A|j' , (9c,d)
and the coefficients C* and C" denote positive- and negative-travelling waves from the origin of
disturbance, respectively. Important remarks on the basic wave propagation characteristics are
summarized from [7]. First, the wave solution of Case III does not exist in the real frequency
space since this type of solution represents a situation in which none of the wave components can
propagate along the waveguide. Therefore the study of Case III is excluded in the present paper.
Second, the vibrating motion of the shaft model in Case I is predominately pure shear [17] which
1296
is unique for the Timoshenko shaft model, while in Case U and Case IV the flexural mode and
the simple shearing mode, which are corrected by including the rotary inertia and shearing effects
in the formulation, dominate. Third, when the shaft rotates at a very high speed and/or the shaft
is axially strained by tensile loads, the wave solution of Case IV governs the vibrating motion of
the shaft model in the low frequency range.
For comparison, the parameters A and B in the simple Euler-Bernoulli beam model are
(10a)
(10b)
where, the non-dimensionalized natural frequency 6) is defined as
is known as the bar velocity).
(10c)
Note that, because B is negative, wave solutions of Case I and Case III do not exist.
In general the displacement and the rotation of an infinitesimal shaft element consist of four
wave components as shown by Eqns. (5a-8b). Once the displacement and the bending slope are
known, the moment M and shear force V at a cross section can be determined from
M = EI^,
dz
(ll)
(du ^
y = J .
(12)
Moreover, the kinematic relationship between the transverse displacement and the slope due to
bending is
S^u 3^u . ,3 r
3r~ 3z^ 3z '
(I3a)
where e' denotes the effects of the axial force and is defined as
e = l + e — .
a
(I3b)
1297
3. WAVE REFLECTION AND TRANSMISSION AT SUPPORTS
When a wave is incident upon a discontinuity, it is transmitted and reflected at different rates
depending on the properties of the discontinuity. Consider a rotating Timoshenko shaft model
supported at ^ = 0 ; see Fig. 2. The support simulates a bearing modelled by linear, translational
and rotational springs, dampers, and a rotor mass which typically represents a gear transmitting a
torque. Based on Eqns. (5a-8b), group the four wave components into 2x1 vectors of positive¬
travelling waves and negative-travelling waves C" , i. e..
Recall that, depending on the system parameters, the rotating Timoshenko shaft model has four
(practically three) different wave solutions in the entire frequency region as described in Eqns.
(5a-8b). Thus C^ and C, in the above expression do not always correspond to propagating and
attenuating wave components, respectively. When a set of positive-travelling waves is
incident upon the support, it gives rise to a set of reflected waves C~ and transmitted waves .
These waves are related by
C'=rC"
(15)
D" = tC",
(16)
where r and t are the 2x2 reflection and transmission matrices respectively and are expressed as
r =
'^u
^12*
(17)
/2!
''22.
’^11
^12
t =
(18)
/2I
^22 _
From Eqns. (5a-8b), suppressing term and excluding Case III, the displacements u~ and
and the bending slopes y/~and y/'^ at the left and right of z = 0, respectively, can be expressed
in terms of the wave amplitudes of the displacement. For convenience, the over-bar (•) on the
wavenumbers is dropped hereafter.
Case 1 {A >0 andB>0);
«-(z) = C, + c;,£'^==, (19a)
W'U) = -n,C:,e-‘^“ -77, c;, +T7,C>-'>''- (19b)
1298
M.Jm
Figure 2. Wave motion at a general support (the disk may be considered as a gear transmitting a torque).
where,
_ 72-®'
T/i — / » ~ / •
r,£ 72^
// ( A > 0 and B < 0 ):
«-(z) = c;, + C'e"''' + ,
i^-(z) = 7,, C, >-'"■= -77,C,-^r''''= +)7,C*e-''“ -TJ,C,;;e"=S
k*(z) = A>“’'" + AV’‘.
where
=
r,£'
> Ba “
CaselV{A<0a.ndB<0):
(19c)
(19d)
(20a, b)
(21a)
(21b)
(21c)
(21d)
(22a, b)
1299
the following set of matrix equations can be established for each Case.
Case / ( A > 0 and 5 > 0 ):
n n
r 1
1 1
. ri n
c^ +
rC^=:
L^i 12]
-^2j
-im, Ic+r IrC"
KYi-tIi) i(r2“7?2)J [-KYi-Vi) -i(.Y2-Tl2)\
(28a)
ri,(k,-J„,co^) + iTi,{c,0)~r,) 0)") + 1772 72 ) ^28b)
ik^~mO)-) + i{c,co + r^-r]^) (fc, -mtt>") + z’(c,® + 72 -772) J
1300
Case // ( A > 0 and S < 0 ):
1
.^1
1
1
L^i
1
^2
tc^
1
1
n
-ir,T7,
[i(r, -T7i) r2-j772_
C -r
-^r, -77,)
~ (r, - it], )_
(29a)
r},(k^-J„^0)-) + irij (c^co-r^)
(k, -mco^) + i(c,Q) + r,-T],)
Tl2(k,-J„,co^-r2) + iV2CrO}
{kj -m(0^ +r2)+/(c,Ct}- 772)
tc",
(29b)
Case /y(A<0and5<0);
'1 11
■ 1 1 ■
rC" =
' 1 r
Jli Hi.
-^2 -^1.
jii ^1-
■-1X2772 -£t,77, ■
C'*
-1X2772
KT.-ni) r, -i77,_
-r
— i(r2 — 772 )
-(r, - in,).
(30a)
T]2ik^-J„^(0'") + iT]^{c,C0-y2)
{k, -m(D‘) + i(c,aj + r2 -772)
riiik,-J„co^-y,) + iT],c^co'
{k, - mco^ + r, ) + i{c,(0 -rij)
tC",
(30b)
where Eqns. (15) and (16) have been applied in all Cases. Note that in Eqn. (27a), it is assumed
that the rotational spring at the support is attached to the cross section of a shaft element such
that the rotational spring responds only to the slope change due to rotation of the cross section
and not the total slope change of the neutral axis of the shaft model. This assumption allows the
shearing motion of the shaft element at the support. Note also that the effect of axial loads on the
shear force at the support is neglected since the contribution of axial loads to the shear force at
the support or boundary is small compared to the shear force due to the flexural motion of the
shaft element. Exact moment and force balance conditions at boundaries for a rotating
Timoshenko shaft element subjected to axial loads can be found in Ref. [16].
The corresponding matrix equations for the simple Euler-Bernoulli shaft model are shown in
Appendix I. Solving the set of matrix equations simultaneously for r and t gives the elements of
the reflection and transmission matrices for each Case. The general forms of solutions to these
sets of equations for each Case is not presented in this paper due to space limitation. However
one can obtain the solutions in either closed-form or numerically. Note that in Case II and Case
IV, the first columns of r and t are the reflection and transmission coefficients due to incident
propagating wave components, and the second columns are due to an incident attenuating wave
component which is generally termed as near-field since this type of wave decays exponentially
with distance. When the distance between the origin of disturbance and the discontinuity is very
1301
large, these attenuating wave components can be neglected. However, as mentioned by many
authors, for example Graff [2], attenuating waves play an important role in wave motions by
contributing a significant amount of energy to the propagating wave components when a set of
propagating and attenuating waves are incident at a discontinuity and, in particular, when the
distances between the discontinuities are relatively small, as in the case of closely-spaced multi¬
span beams. In this paper, near-field components are included. In what follows, the effects of
the point supports on the reflection and transmission of an incident wave are studied. For
comparison, the results are obtained for both the Timoshenko and the simple Euler-Bemoulli
models, which hereafter, for brevity, are denoted by TM and EB, respectively. The system
parameters used in the numerical results are taken from Ref. [10]; ao - 0.0955 m, p = 7700
kgW, K = 0.9, E = 207x10^ Wnf, G = 77.7x10^ N/ml
3.1. Wave reflection and transmission at rigid supports
Consider two cases: the simple support and the clamped support. The r and t are solved and
shown as follows.
• Simple support (k, = oo, = m= c, = - 7„, =0)
Case 7 ( A > 0 and B > 0 ):
r =
_ 1 _
(72 -7i)(r 1/2+6;')
7i(6>"-72)
72(7? -6;-)
7i(6;’-72)
(31a)
t =
_ 1 _
(72-7i)(7i72+6)-)
72(7?-®')
Case 77 ( A > 0 and S < 0):
_ 1 _
_ 1 _
(iT,-r,xr,r,-!V)
Case 7V(A<0andB<0):
_ 1 _
7i(6)"“72)
7i(72 -6)')
r,(r,H®-) 1
r,{r^ + co^)'
(31b)
(32a)
(32b)
(33a)
1302
The corresponding reflection and transmission matrices for the EB model are listed in App. I
Figures 3 and 4 plot the moduli (magnitudes) of the reflection and transmission coefficients
for the simple and clamped supports. The finite cutoff frequencies, above which all waves
propagate, are also marked in the figures. Thus, for the TM model, the wave motions change
from Case 11 to Case 1 when (0>C0^ {0)^-4- in Fig. 3, co, = 4.24 in Fig. 4; O), is slightly altered
by rotation speed and axial load). The results show that, at low frequencies (cD < 0.1 = 3156
rad/sec), the wave reflection and transmission coefficients of the TM model agree well with those
of the EB model for both support conditions. However, as the frequency increases, the wave
propagation characteristics of the TM model differ significantly from those of the EB model.
These differences can be explained by examining the different modes of vibration. When
co>co^ (in the regime of Case 1), the vibrating motion of the TM model is dominated by the pure
shearing motion [7, 17], and hence the EB model, which neglects the rotary inertia and pure
shear effects, become inaccurate at high frequency. As discussed in Ref. [7], at the finite cutoff
frequency, the TM shaft experiences no transverse displacement, and the cross-section of the
shaft simply rotates back and forth in unison
In Figs. 3(d)-(f) and 4(d)-(f), for = 0 and £ = 0 , the reflection and transmission coefficients
of the EB model are independent of the frequency. This is because from Eqn. (10a), A = 0, and
Eqns. (9c, d) lead to a single wavenumber F, = Tj . From Appendix I, Eqns. (32*-36*), the r and
t are thus constant matrices. It is also seen that the wave reflection and transmission coefficients
1303
for both shaft models are basically independent of the rotation speed over the entire frequency
range, even at high rotation speed = 0.05 = 44,600 rpm. In Ref. [7], it is also found that has
negligible effects on the system frequency spectrum, phase velocity and group velocity. On the
other hand, the effects of the axial load are significant for both propagating and attenuating
waves in the regime of Case H, see Figs. 3(b)-(c) and 4(b)-(c). For both shaft models under
simple support and compressive loads (Figs. 3(b, e)), the reflection coefficient ru of the incident
propagating wave is reduced significantly in the regime of Case II, while the transmission
coefficient fj, of the propagating wave component increases to balance the energy carried in the
wave. However, the attenuating wave component which does not carry any energy loses its
transmissibility in the same amount as the reflection coefficient r,2 . Thus, in the presence of a
compressive load, most of the transmitted wave energy in Case II comes from the propagating
component of the incident wave. Note that axial tensile loads have the reverse effects on these
wave components. In the clamped support case, the positive propagating wave component rn is
constant over the regime of Case II under any loading conditions for the both shaft models, as
seen in Fig. 4.
Since there is no damping at the support, the incident power (Ilinc), reflected power (Hrefi) and
transmitted power (Iltnin) in Cases II and IV are related by flinc = rirefl+ritran = Hinc,
or Iriil^+Uiil^ = 1. This relationship is confirmed by the plots shown in Figs. 3 and 4, where for
both shaft models, Irni and knl cannot exceed one. However in the regime of Case I, in which
all wave components propagate, the energy balance is Hinc = (l^ii+r2iP+lfii+r2!p) Ilinc, or Hinc =
(Iri2+r22l^+l?i2+r22l^) Hinc- Together with the plots on the phase of these coefficients (not shown
to minimize the size of this manuscript), the above relationships can also be verified for wave
motion of Case I.
1304
support without “resistance”. The impedance mismatching (rn = 1, ?,, = 0) frequency at which
the propagating wave component is completely reflected without being transmitted can also be
determined from Figs. 4(b, d) for the two shaft models. This impedance mismatching frequency
is located in the regime of Case II for the TM model where the transverse mode dominates the
vibrating motion of the shaft. Numerical results show that, as the spring constant increases, this
impedance mismatching frequency increases, but is limited to within the regime of Case II and
can never be found in the regime of Case I where the pure shearing mode dominates the vibrating
motion of the shaft (refer to Fig. 3 for the transition of types of wave motion).
Figure 6 shows the reflection and transmission coefficients for waves incident upon a support
having both translational and rotational constraints. Since both flexural and shearing modes of
vibration are constrained at this support, the maximum of the reflection coefficient is expected to
be higher than the previous case. Figures 6(a-b) and (c-d) are the results for the TM and EB
models, respectively. The translational and rotational spring constants used in the simulations
are k,Q- 10^ N/m and ^^5= 10^ Nm/rad, respectively. It is noted that in the regime of Case II, i.e..
Figure 6. Wave reflection and transmission coefficients at an elastic support with translational and rotational springs
{k,-k,Q, k^^O, c, = c, = m= y,„ = 0) as a function of frequency, J3 = 0.05 and e = 0. (a-b) and (c-d) are results
for the Timoshenko and Euler-Bernoulli shaft models, respectively.
modulus modulus
2.0
r,2 (upper line)
r,2=:r2, (middle dashed line)
Tj, (lower line)
Figure 4. Wave reflection coefficients at a clamped support (/c, = fc, = ■» and = c, = m = = 0) as a function of
frequency, (a)-(c) and (d)-(f) are the results for the Timoshenko and Euler-BernouUi shaft models, respectively. The
transition from one type of ivave motion to another is marked for the case /J = 0.05 , £ = -0.05 .
3.2, Wave reflection and transmission at elastic supports
Figure 5 shows the reflection and transmission coefficients for waves incident upon a support
with a finite translational spring for three different spring constants. Figures 5(a)-(b) and (c)-(d)
are results for the TM and EB models, respectively. The spring constant used, k^Q = 10^ N/m, is
a typical bearing spring constant value for turbine generators. The plots show that there is no
significant difference in the moduli between the two shaft models. This is because the incident
wave does not experience any rotational constraint at the support, and hence the additional rotary
inertia factor in the TM model has only a small contribution to the wave motions. As the support
spring constant increases, the curves for both the reflection and transmission coefficients are
shifted to the right and, as the spring constant approaches infinity, these curves eventually
become asymptotic to those shown in Fig. 3. Note that an impedance matching (r = 0, t = I),
where all wave components are transmitted without being reflected, is found in the high
frequency region for both shaft models. Thus, as the frequency increases, the characteristics of
waves travelling along the shaft remain unchanged such that waves propagate through this elastic
Figure 5. Wave reflection and transmission coefficients at an elastic support with a translational spring
= c, = c, =m = =0) as a function of frequency, /3 = 0.05 and £ = 0. (a-b) and (c-d) are results for the
Timoshenko and Euler-Bernoulli shaft models, respectively.
1308
in the low frequency range, both shaft models have similar reflection characteristics, and both the
reflection and transmission coefficients are not significantly affected by the rotational spring.
However, as the frequency increases, the effect of the rotational constraint on the wave motion
becomes eminent, particularly for the TM model. As seen in Figs. 6(a) and 6(c), the reflection of
the attenuating wave components are significantly higher than those of the propagating wave
components. Hence, when a rotating shaft has a clamped support(s) such as a journal bearing,
contributions from the attenuating wave components should be included in the formulation since
a significant amount of energy in the propagating component arises from the incident attenuating
wave component. It is noted that the impedance matching regions seen in Figs 5(a, c) disappear
when the rotational constraint is added. Moreover, the impedance mismatching frequency shown
in Figs. 5(b, d), which is found in the regime of Case II, also does not occur. At low frequency in
Figs. 6 (b, d), there appears to be a mismatching region, but rj, is not exactly equal to zero.
From Figs. 6(a, c), it is seen that there is a frequency at which the positive propagating wave
component r,, is zero (this frequency is slightly different for the two models). This frequency
does not correspond to an impedance matching, though the propagating wave is not reflected at
all but is only transmitted (r,i = 1). Based on other research results [18], this phenomenon likely
indicates a structural mode delocalization in bi-coupled systems, in which vibrations on both
sides of the support become strongly coupled. Further research on the vibrations of rotating
shafts with intermediate supports is being pursued to confirm the mode delocalization.
Figure 7 plots the effects of axial compressive loads on the wave reflection and transmission
upon a support with finite spring constant for the Timoshenko shaft model. As seen in Fig. 7(a),
the reflection coefficient for the incident propagating wave component ru is substantially
reduced in the low frequency range while the reflection coefficient for the incident attenuating
wave component increases significantly. However, Fig. 7(b) shows the reversed effects on the
transmission coefficient. It can therefore be concluded that, when the shaft is axially strained by
Figure 7. Wave reflection and transmission coefficients at an elastic support (A:^ = k,Q and kr = c, = Cr = m = J„, = 0)
for the Timoshenko shaft model with and without the compressive load, (a) reflection coefficients, (b) transmission
coefficients.
1309
Figure 8. Wave reflection and transmission coefficients at an elastic support k^ = and c,= Cr = m =
J„, = 0) for the Timoshenko shaft model with and without the compressive load, (a) reflection coefficients, (b)
transmission coefficients.
compressive loads, the energy contribution from the incident attenuating wave component to the
energy in the reflected propagating wave is more significant than the strain-free situation in the
low frequency range, while most of the energy in the transmitted wave derives from the incident
propagating wave component.
Figure 8 plots the wave reflection and transmission coefficients along an axially compressed
Timoshenko shaft model at a support with finite translational and rotational spring constants.
Similar results to the previous example can be observed in terms of energy contribution from the
incident attenuating wave component in the low frequency range. However, the effects of the
axial compressive load on both the reflection and transmission coefficients for the propagating
wave component ( r^^ and r,, ) are significantly reduced when compared to Fig. 7.
3.3. Wave reflection and transmission at damped supports
Figure 9 shows the effects of both translational and rotational dampers at a support with finite
translational and rotational spring constants. Figures 9(a, b) and (c, d) are results for the TM and
EB models, respectively. The translational and rotational damping constants used in this study
are c^to = 2x10^ Ns/m and Cdro = 64x10^ N-m-s/rad, typical values for bearings in turbine
generators. The curves with symbols (• and ♦) are the results when the rotational damping factor
is also included in the formulation. It can be seen that I and Ir^l for both shaft models are
significantly lowered due to the presence of damping. Note that, because of the damping, the
frequency at which 1 rj, I = 0 (compare with Figs. 6(a, c)) no longer exists for both shaft models. It
can also be seen that the effect of the rotational damping factor on the wave reflection and
transmission is not significant over the entire frequency range for both shaft models. For TM
model, the contribution of the rotational damping to both Ir;, 1 and 1?,, I is almost negligible. The
1310
support condition considered in this particular example is simulated as an actual bearing support
adopted in turbine generators. Hence for this particular type of bearing support, the effect of the
rotational damping on wave reflection and transmission is not considerable. Other numerical
results (not shown in this paper) show that the wave propagation at the damped support is
characterized by translational damping rather than rotational damping. Note that similar results
have been presented for the support without damping (see Fig. 6).
Figure 9. Wave reflection and transmission coefficients at an elastic support with damping (k, = k,o, K = Ko, c, = c,;,o,
c, = cjro and m = J,„ = 0) as a function of frequency for /3 = 0.05 and £ = 0. (a-b) and (c-d) are results for the
Timoshenko and Euler-Bernoulli shaft models, respectively.
3.4. Wave reflection and transmission at a rotor mass
Consider a gear rigidly assembled to a rotating shaft. The gear is assumed to be perfectly
balanced and its thickness is sufficiently small such that wave reflection and transmission due to
the geometric discontinuity between the shaft and the gear can be neglected. However the gear
does resist the translational and rotational motions of the cross-sectional element of the shaft.
1311
Figure 10 shows the reflection and transmission upon the gear when the mass mo and mass
moment of inertia J^o of the gear are 4 and 16 times of the shaft, respectively. Not shown in
Figs. 10 (b, d) is that r,2 = 0 when w= /„ = 0. Like some previous support conditions
discussed, the effects of the rotor mass are much more significant in the high frequency region
for both models (particularly around and beyond the cutoff frequency for the TM model). In
general, the rotor mass decreases the transmission and increases the reflection of the wave. At
very high frequency, there is basically no wave transmission. Note that, since the geometric
discontinuity between the shaft and the gear is neglected in this model, one may expect that the
actual reflection for both the propagating and attenuating wave components would be higher.
(a)
(m= = V
(b)
- (m = 0,
- ( m = mg . )
/
Figure 10. Wave reflection and transmission at a rotor mass assembled to a rotating shaft (*, = 0 = L = Cf = Cr = 0,
and m = rriQ, and J,„ = J„^) as a function of frequency when P = 0.05 and £ = 0. (a-b) and (c-d) are results for the
Timoshenko and Euler-Bernoulli shaft models, respectively.
4. WAVE REFLECTION AND TRANSMISSION AT A GEOMETRIC DISCONTINUITY
It is common for a rotating shaft element to have changes in cross-section, or to be joined to
1312
Case I, II, or IV
A, = (1 + a)co- - 2j3co - 16£ (1 + £ - ~)
a
Z=0
Case I, II, or IV
B, = co'
ccco- - 2p(o - I6a (1 + £)(1 + £ - )
a
A={\ + a)(o^-2^co-^^^ (l + £, -— )
o" a
16a
= co^\ a (O' -215(0-^^ (l+£j(l + £ -— )
L O’ a
Figure 11. Wave reflection and transmission at a geometric discontinuity.
another shaft element by a coupling. Figure 1 1 shows a typical example of a discontinuous shaft
model in which two shafts of differing wavenumber and diameter are joined at z = 0 . The
subscripts I and r denote z = O' and z = 0^ regions, respectively. It is known that when a wave
encounters a junction or a discontinuity, its wavenumber is changed. It is therefore possible that
a wave on the left side of the junction can be propagating, while after crossing the junction to the
right side, the wave becomes attenuating. Therefore, for a Timoshenko shaft, when a wave
propagates through the junction, there are mathematically nine possible different combinations of
wave motions to be considered depending on the values of the functions A and B on each side of
the junction, as depicted in Fig. 12.
Figure 12. Nine possible combinations of wave motions at a geometric discontinuity of the cross section for the
Timoshenko shaft model. Subscripts / and r denote the left and the right side of the discontinuity, respectively.
1313
For simplicity, assume that material properties such as p, E, and G are the same for both sides
of shaft element. The displacement continuity, moment and force equilibrium conditions are
applied at the junction to determine the wave refection and transmission matrices. Results for the
three most commonly encountered possibilities in the low frequency regime are listed as follows.
Case II {A, >0, Bi <0) - Case I (A^ > 0 , R, >0):
■ 1 r
1
1 '
r 1 M
rC" =
tCY
Jlu ^21 .
nu ■
~n2i.
Jl\r ^2r_
r -i^Bu
^21^2!
C" +
r Xtnu
”Y,r72, 1
L^xr.,
-riu) r
21 -in 21^
1
Y
rr
1
T
^772,)J
rC-^
Yuriu 72rn2r
L-io-"(r„-77„) -i<y^(r2r-V2r)J
tc",
Case II (A, >0, B, <0) - Case II (A^ > 0 , B^<0):
■ 1 1 ■
■ 1 1 ■
■ 1 1 *
C" +
rC^ =
Jlv n2l.
— nu “^2/.
Y.r n2r.
(37a)
(37b)
(38a)
-"^21^21
^2, -in 21
C" +
i^unu ^21^21
~i(^H — nu ) ~(^2/ ~ ^^21 )
rC"
-i<yXrnu -^Xr'n2r
-zcr"(r,,-77„) -C7“(r2,-7]2,)J
CaseII{A^>0, B, <0) - Case IV {A, <0 , S, <0):
r i 1 1
r 1 1 1
■ 1 r
C" +
rC^ =
nu n2i
l-nu -n2i^
n2r n^r.
Y/^2/
^2/^2/
JXi—nu) Y/~^*^2;.
“(Yf -in2i)_
-i^%rn2r
-<yXrn:r
tc\
(38b)
(39a)
(39b)
where cr is the diameter ratio between the shaft elements, defined as
1314
(40)
CJ = — .
Note that 77/5 in Eqns. (37a-39b) are given by Eqns. (20a, b), (22a, b), and (24a, b) according to
the type of wave motion, and 77/ s on the right side of the geometric discontinuity are modified
as follows.
where
: - 77 = - — for Case I,
r„e; yzX
(41a, b)
-co^ Tl+co-
, Ti, - — - 7- for Case II,
r„e; iT^rS;
(42a, b)
,7]2 for Case IV,
ir,x r,,e;
(43a, b)
£' = 1 + 8,. — and £, = — .
(44)
a O'
Moreover, the wavenumbers, A and B of the shaft element on the right side of the junction are
modified as follows.
y „ = ^[a, + ^Aj-4B^f . r,, = - Va?-4B,)^ , (45a. b)
+ = . (45c, d)
where,
A. = (l + a)a.= -2pa)-^a + £,-|),
B. =co‘
am--2Pa-^ (l + £,)(l + £,-^)
C OL
(46b)
Corresponding results for the simple Euler-Bemoulli shaft model are listed in Appendix II.
Figures 13 to 16 show some representative examples of wave reflection and transmission
upon the geometric discontinuity. In Figs. 13 and 14, the thick and thin curves represent results
1315
for the TM and EB models, respectively. The second graph in each figure shows the changes of
Ai, Bi, Ar and Br, and how wave solutions on both sides of the discontinuity change as the
frequency increases for the TM model. In general, the wave reflection and transmission for the
EB model are frequency independent except when the shaft is axially strained, while the wave
propagation characteristics for the TM model are strongly dependent on the frequency.
Comparing Figs. 13 and 14, it is noted that, for both shaft models, the average reflection and
transmission rates for cr = 0.8 are higher than those for o' = 1.2 , especially for the attenuating
wave components. These results imply that incident attenuating waves contribute more energy to
propagating waves at the discontinuity when the waves travel from a smaller to a larger cross-
section. In particular, it is noted that the transmissibiiity of the attenuating wave tn has a strong
dependency on the direction of propagation. Note also that the differences between the two shaft
models are more pronounced when cr = 0.8 . It is clearly seen from the figures that when Bj and
B^ change from negative to positive, both reflection and transmission coefficients experience a
sharp jump or drop at the finite cutoff frequencies, due to changes in the types of wave motion.
In the frequency region (S, > 0 and B^ <0) located between the two cutoff frequencies in Fig.
2.0 h
0.5
Thick curves : Timoshenko shaft model
Thin curves : Euier-Bernoulli shaft model
- Ui
h
i
il _
-
7/ X. - V'
.'7
\
5
(O
Figure 13. Reflection and transmission of waves incident upon a change in the cross-section, a= 0.8, P = 0.05, and
£ = 0. Thick and thin curves are results for the Timoshenko and Euier-Bernoulli shaft models, respectively. Note
that the ordinates in the lower graphs keep increasing with frequency (abscissa).
1316
13, the wave motion on the left side of the junction is governed by the wave solution of Case I
since all wave components are propagating at a frequency larger than the cutoff frequency, while
the wave motion on the right side of the junction is governed by the wave solution of Case II.
Thus, for O’ = 0.8 , some of the propagating wave components on the left side of the shaft element
cannot propagate as they pass the discontinuity, and become attenuating. A similar, but converse
conclusion can be drawn for the frequency region (5^ > 0, 5, < 0) when cr = 1.2 , as shown in
Fig. 14. The results of Figs. 13 and 14 show that, for different system parameters cr, jS, and e and
at any given frequency, the types of wave motion on each side of the discontinuity can be
different, as depicted in Fig. 12.
From Eqns. (41a-43b), it is seen that when the Timoshenko shaft is axially strained and (O is
not sufficiently large, the wavenumber (hence wave propagation characteristics) depends strongly
on the cross-section ratio <7. Figure 15 shows the effects of the axial load on the wave reflection
and transmission, which are mostly limited to the relatively low frequency region. In Figs. 15(a-
b), when the shaft is axially compressed (£ = -0.05), the reflection and transmission due to the
incident attenuating wave component decrease for both o < 1 (plot (a)) and <7 > 1 (plot (b)).
However, the transmission due to an incident propagating wave decreases significantly for
— — f, 2
2.0 h
Figure 14. Reflection and transmission of waves incident upon a change in the cross-section, cr= 1.2, /3 = 0.05, and
£ = 0. Thick and thin curves are results for the Timoshenko and Euler-Bernoulli shaft models, respectively. Note
that the ordinates in the lower graphs keep increasing with frequency (abscissa).
1317
Figure 15. Reflection and transmission of waves upon a change in the cross-section when = 0.05 for the
Timoshenko shaft model, (a) <7=0.8 and £ = -0.05. (b) (T= 1.2 and e= -0.05. (c) cr= 0.8 and £= 0.05. (d) or = 1.2
and e= 0.05. Thin and thick curves show the results when the shaft is strain-free (£= 0) and strained, respectively.
c - 0.8 and increases for <7 = 1.2 at low frequency.
Effects of the axial load on the wave reflection and transmission are more significant when
the shaft is compressed (Figs. 15(a-b)) than when it is under tension (Figs. 15(c-d)). This is
because the wavenumbers of both the propagating and attenuating wave components are only
slightly changed. It is also noted that, in the low frequency range, the wave solution of Case IV
governs the wave motions on both sides of the discontinuity, and the wave components which
have large wavenumber (Fi) attenuate, while wave components with small wavenumber (r2)
propagate along the waveguide as long as A remains negative.
1318
5. WAVE REFLECTION AT BOUNDARIES
When a wave is incident upon a boundary, it is only reflected because no waveguide exists
beyond the boundary. Consider an arbitrary boundary condition with translational and rotational
spring constraints, dampers, and a rotor mass, as shown in Fig. 16. The reflection matrix at the
boundary is derived for each Case. Applying the same non-dimensional parameters employed in
Section 3, and by imposing the force and moment balances at the boundary, which can be
deduced by eliminating and inEqns. (27a, b),
M~ = k^y/ + c^yr + J„y/ , (47 a)
-V = k,u + c,u + mu , (47b)
the reflection matrix for each Case is determined.
CaseI{A>0, B>0):
J r72(ir2-2:,J T’r-T7,(jri + ^.) +
+ Kr2-Bi) + ^s\ Kr2-T72)-^.J’
Case 7/ ( A > 0 , 5 < 0):
7],(ir,-2:j T'r-77,(ir,+Ej -7]2(r2 + zj1
'''[i(r,-r7,)+z, (r,-ii7,)+i.J [i(r,-j?,)-z,
M,Jm
Z=0
Figure 16. Wave reflection upon a general boundary.
1319
CaseIV{A<0, 5<0):
772 (^r2 s„,)
iCFj - 7)2) +
(r.-mj+s.
772 (zTj + Sp,)
/(r2 ~ ^2) ■“
-^iCr.+zj
(r,-j77,)-E,
(50)
where 77's in above equations have been defined in Eqns. (20a, b), (22a, b) and (24a, b), and
Z„, = + ic^co - J,„co~ , and = ^, + ic,Q) - mco^ , (51a, b)
The corresponding results for the simple Euler-Bernoulli shaft model are listed in the Appendix
in. By specifying the parameters in the reflection matrix r, results for three typical boundary
conditions (simple support, clamped support, and free end) can be obtained.
• Simple support {k, =^, =m = c, = = 7,„ = 0)
r
-1 0
0 -1
for Case I, II, and IV,
(52)
• Clamped support (k^ = k,. = m = c, = = J„, = 0)
1
Tli-ri2
(53b)
(53c)
• Free end {k, = k^ = m = c, = = /„, = 0 , and £ = 0)
1
rii+ri2
2772
77,- 772 L-277i -(r?, +772)J
1
-irii+Tli) "2772
2t}, (771+772)
for Case II,
for Case IV,
■77, +7?2 2772
-277, -(771+772),
for Case I,
(53a)
r
J_
77,7?2(ri+72)”7ir2(77, +772)
-277,7i(77, -7i)
277272(772-72)
-77,772(71 +72) + 7i72(77i +772).
(54a)
where. A, = r],n, - 7, ) + 7,72 (77, - ^2 ) for Case I,
+72) + 7i72(77i +772) 2i7}^y^iin^ -y^)
^ A;, [ 2/77,7,(771 -7,) 77,772(/7, +72)-7i72(7?, +772)
(54b)
where, =77,772 (77, -72) -7,72(7?, - 772) for Case//,
1320
Figure 17. An example of a rotating shaft with multiple supports and discontinuities.
I \TiiV2(ri+ir2)-rj2(Tii+r]2) 277,7,(77, +ir,) ' ^
r = - (54c)
A/v L -2i7?2r,(77j-7j -77,772(7, +J72)+rir2(^i +^2).
where, A;^ =77,772(71 -172) “7172 (Hi -772) for Case IV.
6. APPLICATIONS
The reflection and transmission matrices for waves incident upon a general point support or a
change in cross-section can be combined with the transfer matrix method to analyze the free
vibration of a rotating Timoshenko shaft with multiple supports and discontinuities, and general
boundary conditions. The basic idea of this technique has been shown in Ref. [6]. However, due
to the complex wave motions in the Timoshenko shaft model, such as the frequency dependency
of the wave reflection and transmission at a cross-section change, it is important to apply the
proper reflection and transmission matrices consistent with the values of A and B on both sides of
the discontinuity, particularly when numerical calculations are performed. Consider for example
the free vibration problem of the rotating Timoshenko shaft model shown in Fig. 17. Denoting R
as a reflection matrix which relates the amplitudes of negative and positive travelling waves at a
discontinuity, and defining T/ as the field transfer matrix which relates the wave amplitudes by
C^iZo + z) = TC"(Zo) , C-(zo + z) = T-'C-(zo) , (55)
the following relations can be found.
1321
W-=R5W^
(R5=>-s).
(56a)
i = 2,3,4 (station number),
/i = left (/) or right (r)
(56b)
w>T,w-,,
(56c)
< =r,w-,
(56d)
<=T.w;,
(56e)
where in Eqn. (56b),
R,,=TiR„,,T,. R,,=r,+t,(R-’-r,-)-'t, (56f)
Solving the above matrix equations gives
(r,T,R„T,-I)w>0, (57)
where each element of the matrix is a function of two different wavenumbers and the frequency
CO. For non-trivial solutions, the natural frequencies are obtained from the characteristic equation
Det[(r,T,R„T,-I)] = 0. (58)
The proposed method is applied to an example of a two-span rotating shaft, simply supported
at the ends and with an intermediate support consisting of translational and rotational springs, as
shown in Fig. 18. Numerical computations were performed by a PC-based Mathematical. The
values of the spring constants Kj and Kr are those introduced in Section 3.2, with ^ = 1 m and
the rotation parameter p = 0.05 .
1322
Figure 19 shows the first eight natural frequencies of the vibrating shaft for both the TM and
EB models for a classical simple intermediate fixed support K^=0) placed at
various locations. The results confirm the well-known fact that the Timoshenko model leads to
smaller eigenvalues. Figure 20 shows the first eight natural frequencies of the Timoshenko shaft
for an elastic intermediate support with three different translational spring constants. It can be
seen that the effect of the translational spring diminishes for higher modes. The proposed wave
analysis technique can also be applied effectively to the study of structural mode localizations in
mistuned, rotating systems. Dynamics of such systems will be addressed in another paper.
Figure 19. Natural frequencies of a two-span, rotating
shaft as a function of the support location; intermediate
support is fixed.
Figure 20. Natural frequencies of a two-span rotating
Timoshenko shaft as a function of the support location;
intermediate support consists of k, and kr .
7. SUMMARY AND CONCLUSIONS
In modern high speed rotating shaft applications, it is common that the shaft has multiple
intermediate supports and discontinuities such as bearings, rotor masses, and changes in cross-
1323
sections. In many cases, the ratio of the shaft diameter to its length between consecutive supports
is large, and the Timoshenko model (TM) is needed to accurately account for the shear and rotary
inertia effects. In this paper, the wave propagation in a rotating, axially strained Timoshenko
shaft model with multiple discontinuities is examined. The effect of the static axial deformation
due to an axial load is also included in the model. Based on results from Ref. [7], there are four
possible types of wave motions {Cases I, II, III and IV) in the Timoshenko shaft, as shown by
Eqns. (5a-8b). In practice. Case III does not occur and is excluded in the analysis. For each
Case, the wave reflection and transmission matrices are derived for a shaft under various support
and boundary conditions. Results are compared with those obtained by using the simple Euler-
Bemoulli model (EB) and are summarized as follows.
1) In general, the two shaft models show good agreement in the low frequency range where the
wave motion is governed by Case II and Case IV. However, at high frequencies, the types of
wave motions and propagation characteristics for the TM and EB models are very different.
2) The effects of shaft rotation on the wave reflection and transmission are negligible over the
entire frequency range and even at high speed (up to 44,600 rpm). While the effects of the
axial load are significant, especially in the low frequency range.
3) When waves are incident at supports with only translational springs, differences in the results
between the TM and EB models are small, and there exists frequency regions of impedance
matching and an impedance mismatching frequency (limited to within the regime of Case IT).
The impedance matching and mismatching disappear when a rotational spring is added to the
support. Instead, there is a frequency at which Irul = 0 and Ifni = 1, and vibrations on both
sides of the support become strongly coupled. This (delocalization) phenomenon suggests
further research on the vibrations of constrained multi-span beams. When there is damping
at the support, the frequency at which IrnI = 0 does not occur. Moreover, effects of
translational damping on the wave propagation are more significant at high frequency,
especially for the TM model, however effects of rotational damping is not significant over the
entire frequency range.
4) Contributions of attenuating wave components to the energy in the reflected and transmitted
waves are significant when the shaft is axially strained and when the support has a rotational
constraint. Thus attenuating waves should be included in the formulation.
5) Unlike the spring supports, in which waves are easily transmitted at high frequency, the rotor
mass support diminishes the wave transmission as the frequency increases.
6) When waves are incident at a geometric discontinuity such as a change in the cross-section,
there are nine possible combinations of wave motions on both sides of the discontinuity. It is
shown that differences of the results between the TM and EB models depend on the diameter
ratio (and hence the direction of the wave incidence). Moreover, incident attenuating waves
contribute more energy to propagating waves at the discontinuity when the waves travel from
a smaller to a larger cross-section. When the shaft is axially strained, the effects of the load
on the wave propagation are primarily limited to the low frequency range.
The reflection and transmission matrices are combined with the transfer matrix method to
provide a systematic solution method to analyze the free vibration of a multi-span, rotating shaft.
Since the procedure involves only 2x2 matrices (while including the near-field effects already),
strenuous computations associated with large-order matrices are eliminated.
1324
ACKNOWLEDGMENTS
The authors wish to acknowledge the support of the National Science Foundation and the
Institute of Manufacturing Research of Wayne State University for this research work.
REFERENCES
1. Lin, Y.K., Free Vibrations of a Continuous Beam on Elastic Supports. International Journal
of Mechanical Sciences, 1962, 4, pp. 409-423.
2. Graff, K.F., Wave Motion in Elastic Solids, Ohio State University Press, 1975.
3. Cremer, L,, Heckl, M. and Ungar E.E., Structure-Bome Sound, Springer-Verlag, Berlin,
1973.
4. Fahy, F., Sound and Structural Vibration, Academic Press, 1985.
5. Mead, D.J., Waves and Modes in Finite Beams: Application of the Phase-Closure Principle.
Journal of Sound and Vibration, 1994, 171, pp. 695-702.
6. Mace, B.R., Wave Reflection and Transmission in Beams. Journal of Sound and Vibration,
1984, 97, pp. 237-246.
7. Kang, B. and Tan, C.A., Elastic Wave Motions in an Axially Strained, Infinitely Long
Rotating Timoshenko Shaft. Journal of Sound and Vibration (submitted), 1997.
8. Han, R.P.S. and Zu, J.W.-Z., Modal Analysis of Rotating Shafts: A Body-Fixed Axis
Formulation Approach. Journal of Sound and Vibration, 1992, 156, pp. 1-16.
9. Zu, J.W.-Z. and Han, R.P.S. , Natural Frequencies and Normal Modes of a Spinning
Timoshenko Beam With General Boundary Conditions. Transactions of the American
Society of Mechanical Engineers, Journal of Applied Mechanics, 1992, 59, pp. 197-204.
10. Katz, R., Lee, C.W., Ulsoy, A.G. and Scott, R.A., The Dynamic Response of a Rotating
Shaft Subject to a Moving Load. Journal of Sound and Vibration, 1988, 122, pp. 131-148.
11. Tan, C.A. and Kuang, W., Vibration of a Rotating Discontinuous Shaft by the Distributed
Transfer Function Method. Journal of Sound and Vibration, 1995, 183, pp. 451-474.
12. Argento, A. and Scott, R.A., Elastic Wave Propagation in a Timoshenko Beam Spinning
about Its Longitudinal Axis. Wave Motion, 1995, 21, pp. 67-74.
13. Dimentberg, F.M., Flexural Vibrations of Rotating Shafts, Butterworth, London, 1961.
14. Dimarogonas, A.D. and Paipeties, S.A., Analytical Method in Rotor Dynamics, Applied
Science, New York, 1983.
15. Lee, C.W., Vibration Analysis of Rotors, Kluwer Academic Publishers, 1993.
16. Choi, S.H., Pierre, C. and Ulsoy, A.G., Consistent Modeling of Rotating Timoshenko Shafts
Subject to Axial Loads. Journal of Vibration and Acoustics, 1992, 114, pp. 249-259.
17. Bhashyam, G.R. and Prathap, G., The Second Frequency Spectrum of Timoshenko Beams.
1325
Journal of Sound and Vibration, 1981, 76, pp. 407-420.
18. Riedel, C.H. and Tan, C. A., Mode Localization and Delocalization of Constrained Strings
and Beams. Proceedings ofASME Biennial Conference on Mechanical Vibration and Noise
(submitted), 1997
1326
For simple and clamped supports, the reflection and transmission matrices are listed as follows.
Simple support ( k. = k. = n
l = C,
= c, = .
Case // ( A > 0 , B < 0):
1
■ r.
r, ■
iF, r2
'ir,
r./
'"ir.-r2
.-ir^
-r^.
CaseIV{A<0, B<0):
1
-ire
1 1
t “
'r.
-ir:
iT, +r2
.-^2
-n_
i ^T,+r2
-r2
• Clamped support {k^ =<=<>, m=c, = c, = J„, = 0 ); t = 0 .
CaseII{A>0, B<0):
1 ^^1+^2 2r2
-2iT, -(ir, + r,)J’
CaseIV{A<0, B<0):
1 r-(ir,-r2) -2iTi '
^“iT.+r.L -2r2 iT, -r^ ■
(32a*, b*)
(33a*, b*)
(35*)
(36*)
APPENDIX II
The reflection and transmission matrices for a wave incident upon a cross-sectional change
for the simple Euler-Bernoulli shaft model can be determined by solving the following sets of
matrix equations. Only two representative combinations are shown.
CaseII(Ai>0, B, <0)- Case II {A,>0, B,<0):
1327
Casen{A,>0, <Q) - Case TV {A, <0 , <0):
1
1
1
1 1 .
1
1 1 .
■ +
-
•^2,.
-F^ F"
^1/ •*■2/
C*
+
F2 1
rC" =
[T^ _r3
L “ 1/ ^ 21
2/ J
r,rj
where, F sr and T2r have been defined in Eqns. (45c, d), and A^- and Br are given by
O’"
G
(39a')
(39b')
(46a')
(46b*)
If the rotating shaft is strain-free, then r can be reduced to simple forms representing typical
boundary conditions such as simple support, clamped support, and free end as shown in Ref. [6].
Note that for those supports in the strain-free case, the reflection matrices are constant.
1328
ANALYTICAL MODELLING OF COUPLED VIBRATIONS OF
ELASTICALLY SUPPORTED CHANNELS
Yavuz YAMAN
Department of Aeronautical Engineering, Middle East Technical University
0653 1 Ankara, Turkey
An exact analytical method is presented for the analysis of forced vibrations
of uniform thickness, open-section channels which are elastically supported at
their ends. The centroids and the shear centers of the channel cross-sections do
not coincide; hence the flexural and the torsional vibrations are coupled. Ends
of the channels are constrained with springs which provide finite transverse,
rotational and torsional stiffnesses. During the analysis, excitation is taken in
the form of a point harmonic force and the channels are assumed to be of type
Euler-Bernoulli beam with St.Venant torsion and torsional warping stiffness.
The study uses the wave propagation approach in constructing the analytical
model. Both uncoupled and double coupling analyses are performed. Various
response and mode shape curves are presented.
1. INTRODUCTION
Open-section channels are widely used in aeronautical structures as stiffeners.
These are usually made of beams in which the centroids of the cross-section
and the shear centers do not coincide. This, inevitably leads to the coupling of
possible flexural and torsional vibrations. If the channels are symmetric with
respect to an axis, the flexural vibrations in one direction and the torsional
vibrations are coupled. The flexural vibrations in mutually perpendicular
direction occur independently. In the context of this study, this type of
coupling is referred to as double-coupling. If there is no cross-sectional
symmetry, all the flexural and torsional vibrations are coupled. This is called
as triple-coupling. The coupling mechanism alters the otherwise uncoupled
response characteristics of the structure to a great extent.
This problem have intrigued the scientists for long time. Gere et al [1], Lin
[2], Dokumaci [3] and Bishop et al [4] developed exact analytical models for
the determination of coupled vibration characteristics. All those works, though
pioneering in nature, basically aimed to determine the free vibration
characteristics of open-section channels.
The method proposed by Cremer et al [5] allowed the determination of
forced vibration characteristics, provided that the structure is uniform in
cross-section. The use of that method was found to be extremely useful when
the responses of uniform structures to point harmonic forces or line harmonic
loads were calculated. Mead and Yaman presented analytical models for the
1329
analysis of forced vibrations of Euler-Bernoulli beams [6]. In that they
considered finite length beams , being periodic or non-periodic, and studied
the effects of various classical or non-classical boundary conditions on the
flexural response. Yaman in [7] developed mathematical models for the
analysis of the infinite and periodic beams, periodic or non-periodic Kirchoff
plates and three-layered, highly damped sandwich plates.
Yaman in [8] also developed analytical models for the coupled vibration
analysis of doubly and triply coupled channels having classical end boundary
conditions. In that the coupled vibration characteristics are expressed in
terms of the coupled wave numbers of the structures. The structures are first
assumed to be infinite in length, and hence the displacements due to external
forcing(s) are formulated. The displacements due to the waves reflected from
the ends of the finite structure are also separately determined. Through the
superposition of these two, a displacement field is proposed. The application
of the end boundary conditions gives the unknowns of the model. The
analytical method yields a matrix equation of unknowns which is to be solved
numerically. The order of the matrix equation varies depending on the number
of coupled waves. If the cross-section is symmetric with respect to an axis
( double-coupling) and if the warping constraint is neglected, the order is six.
If there is no cross-sectional symmetry (triple-coupling) and if one also
includes the effects of warping constraint, the order then becomes twelve.
This order is independent of the number of externally applied point forces.
Although the method is basically intended to calculate the forced response
characteristics, it conveniently allows the computation of free vibration
characteristics as well. The velocity or acceleration of a point can easily be
found. The mode shapes can also be determined. Both undamped and damped
analyses can be undertaken.
This study is based on the models developed in reference [8] and aims to
analyze the effects of non-classical end boundary conditions on the coupled
vibratory responses. If the ends are elastically supported (which may also have
inertial properties) the problem becomes so tedious to tackle through the
means of classical analytical approaches. The current method alleviates the
difficulties encountered in the consideration of complex end boundary
conditions.
In this study a typical channel, assumed to be of type Euler-Bernoulli beam, is
analyzed. It represents the double-coupling. Effects of the elastic end
boundary conditions on the resonance frequencies, response levels and mode
shapes are analyzed. Characteristics of otherwise uncoupled vibrations are
also shown.
1330
2. THEORY
2.1 Flexural Wave Propagation in Uniform Euler-Bernoulli Beams
Consider a uniform Euler-Bernoulli beam of length L which is subjected to a
harmonically varying point force Fo e acting at x=Xf. The total flexural
displacement of the beam at any Xr (0 < Xr < L) can be found to be [5-8],
w(x,,t) = ( i;A„e''„\ + Fo i ane'^'
r r
)e'
(1)
The first series of the equation represents the effects of four waves which are
being reflected from the ends of the finite beam. They are called free-waves.
The second series accounts for the waves which are being created by the
application of the external force Fo e on the infinite beam. Those waves are
known as forced-waves, kn is the n’th wave number of the beam and
kn =(mco^/EI)'^'‘ where m= Mass per unit length of the beam, co= Angular
frequency, EI= Flexural rigidity of the beam, an values are the complex
coefficients which are to be found by satisfying the relevant compatibiliy and
continuity conditions at the point of application of the harmonic force [6,7].
An values, on the other hand are the complex amplitudes of the free waves and
are found by satisfying the required boundary conditions at the ends of the
beam. Once determined, their substitution to equation (1) yields the flexural
displacement at any point on the finite beam due to a transversely applied
point harmonic force. More comprehensive information can be found in [7].
2.2 Torsional Wave Propagation in Uniform Bars
If one requires to determine the torsional displacements generated by a point,
harmonically varying torque, a similar approach to the one given in Section
2.1 can be used. In that case, the total torsional displacement can be written as:
= )e‘”' (2)
k is the wave number of the purely torsional wave and is known to be
k=(-pIoa)VGJ)^^^ . k2= -ki and GJ=Torsional rigidity of the beam, p=Material
density, Io=Polar second moment of area of the cross-section with respect to
the shear centre. Toe'“ ^ is the external harmonic torque applied at x=Xt and
b=l/(2kGJ). Bn values are the complex amplitudes of the torsional free-waves
and are found by satisfying the appropriate end torsional boundary conditions.
The consideration of the warping constraint To modifies equation (2) to the
following form.
1331
(3)
«x„t) = (£C„e^\ + Toic„e-'=„'V,' ) e
n*l »•!
Now kn are the roots of
EFo kn'^-GJkn^-pIoCO^=0
(4)
Cn values are found by satisfying the necessary equilibrium and compatibility
conditions at the point of application of the point harmonic torque acting on an
infinite bar [8]. Cn values are determined from the end torsional boundary
conditions of the finite bar.
2.3 End Boundary Conditions for Uncoupled Vibrations
2.3.1 Purely Flexural Vibrations
Consider an Euler-Bernoulli beam of length L which is supported by springs at
its ends. The springs provide finite transverse and rotational constraints Kt and
Kr respectively. The elastic end boundary conditions can be foimd to be:
El w”(0) - Kr I w’(0)=0 El w’”(0) + Ktj w(0)=0
El w”(L) + K r,r w’(L)=0 El w’”(L) - Kt,rW(L)=0 (5)
Here w’=
dw(x)
w
d‘w(x)
and w’”
dV(x)
w(x) is the spatially
dx ’ dx“ dx'
dependent part of equation (1) and second subscripts 1 and r allows one to use
different stiffnesses for left and right ends. A more comprehensive study on
these aspects can be found in references [6,7].
2.3.2 Purely Torsional Vibrations
Now consider a bar of length L which is supported by torsional springs,
having finite Ktor,at its ends. The elastic end boundary conditions requires that,
Torque (0) - Ktor,i (|)(0)=0 and Torque (L) + Ktor .r (i)(L)=0 (6)
Depending on the consideration of the warping constraint To, the torque has
the following forms
Torque(x)=GJ — — or Torque(x)= GJ — — - Ei o , 3 ( /)
dx dx ux
1332
(})(x) in equation (7) should be obtained either from equation (2) or equation (3)
depending on the warping constraint r©.
2.4 Doubly-Coupled Vibrations
Now, consider Figure 1 . It defines a typical open cross-section which is
synunetric with respect to y axis
V z
(b)
Figure 1 : A Typical Cross-section of Double-coupling
( a. Coordinate System, b. Real and Effective Loadings
C: Centroid, O: Shear Centre)
A transverse load applied through C results in a transverse load through O and
a twisting torque about O. In this case the flexural vibrations in z direction are
coupled with the torsional vibrations whereas the flexural vibrations in y
direction occur independently. The motion equation of the coupled vibrations
is known to be [1,2].
a-w a-(j)
= 0
a‘‘(i) 3^6 a’w a"(j)
(8)
If one assumes that,
w(x,t) = w„eVe”'
(|)(x.t) = 4>„eVe“’ (9)
1333
Then, it can be found that, a load Pz through the centroid will create the
following displacements at any x (0 < x < L) along the length of the
channel [8],
w(x,t) = (2A„e^’‘+Pjt a„ e'^ )e'"'
n^l nvl
n«l n>I
Now kn values are the coupled wave numbers, An values are the complex
amplitudes of the coupled free waves, an values are the complex coefficients
which are to be found by satisfying the required compatibility and continuity
conditions and 'Pn^ ( (El^ kn'^-mco^) / (CymcD^) ) [8].
If required, the warping displacement u(x,t) can be found from (l)(x,t )as
u(x,t)=-2A,^iM (11)
dx
where As is the swept area.
Here 2j gives the order of the motion equation. j=3 defines the case in which
the effects of warping constraint are neglected and j=4 represents the case
which includes the warping effects.
An values are found by satisfying the necessary 2j end boundary conditions. If
warping constraint is neglected, the required six boundary conditions have the
general forms given in equations (5) and (6). But the forms of w(x) and (j)(x)
are now those given by equations (10) with j=3. If the warping constraint is
included in the analysis the boundary conditions become eight. The six of
those are again found by considering equations (10) with j=4 and substituting
the resultant forms into equations (5) and (6). The remaining two can be
found by evaluating equation (1 1) at both ends.
When the flexural and torsional displacement expressions are substituted into
the relevant equations, a set of equations is obtained. For the case of a load
Pz and no warping constraint, the following equations can be found for j=3.
EI^ w’”(0) + Kt,iw(0)=0
EI^ (E kn^ An+ (-1) PzZ - kn^ a „ 6 ' "f ' )
^t,l ( E ■^n ■^PzE^n^ n f ) “0 (12)
1334
(13)
EI^w”(0) -Kr,i w’(0)=0:
EI^( I kn'An + Pz t kn'an 6 ‘ ' )
n«l n«t
■ ( S kn An + Pz^'kn^n® n f )
n*l n“l
GJ^^l«=o-K,„,.i W)=0:
dx
GJ( S k„ T „ A„ + (- 1 ) (Pz cy) t -k„ 'f „ a„ e ' ’‘f ')
fl«l 11' I
-K,or,i ( i % A„ + (Pz Cy) i >?„ a„ e ')=0
diKx) I
GJ-^Ix.l + K,„„ <KL)=0:
dx
GJ( 2k„>P„ A„ e“+ (PzCy) i-k„'P„a„ )
ns I 11=1
+ K,„,.r ( I % A„ + (Pz Cy) t Tn a„ e "f ')=0
El5W”(L)+ Kr,rW’(L)=0:
El^CS k„^A„e^‘-+(Pz) i k„^a„ )
n-l ii’*)
+ Kz,r( i k„A„ e“+ (Pz) t-k„a„ e■^'‘-^')=0
n»l n=i
(14)
(15)
(16)
EI^ w’”(L)-Kt,rW(L)=0:
Eiaz kn'Ane'n^ +(Pz) 2 - kn^ a „ e n ' ^ )
11=1 Iia]
-K,|(Z A„eV+(Pz) Za„e^''-V)=0 (17)
lt=l 11=1
Here (-1) multipliers are included due to the symmetry and anti-symmetry
effects.
Those equations can be cast into the following matrix form.
= -{Terms containing Pz} (18)
1335
An eighth order equation represents the necessary matrix equation for the
determination of An values if the warping constraint is included in the analysis.
In that case, equations (12), (13), (16) and (17) are valid with j=4. On the other
hand equations (14) and (15) should be replaced by,
(GJ^-Er„^^)L.o <l>(0)=0:
dx dx
(GJ( £ k„ % A„ + (- 1 ) (P. Cy) £ -k„ a„ e * „ ' ’‘f ') -
nol n»l
Er„( £ k„3 'f „ A„ + (- 1 ) (P^ Cy) £ -k„ = % a„ e * „ ' -f '))
it«l IIS’!
-K,„,,,(|;'P„A„+(P,Cy)i 'P„a„e-^'’‘f')=0 (19)
cr A/T^=A.
(GJ “ EFo j 3 )• x=L I^tor>r y(L) 0 .
dx dx
(GJ( Jkn'Fn A„ e’=„‘-+ (PjCy) i-kn'Pnan ' )-
n=t Ii»l
EFoC Xkn^ 'J'n A„ eV+ (PzCy) £-k„^'P„a„ ' ))
11=11 n-l
+ Kior,, ( X % A„ + (Pz Cy) X % a„ e * „ ' \ ')=0 (20)
n-] ii3|
where j=4. The remaining two equations are found by considering the warping
of the extreme ends. If the ends are free to warp the axial stress is zero, if the
ends are not to warp the axial displacements are zero at both ends. No elastic
constraints are imposed on end warping. If the left end is free to warp and the
right end is not to warp, the required boundary conditions can be shown to be;
u’(0)=0:(x kn^'Pn A„ + PzX 'i'.ian k„^e-^ ' ^') = 0 (21)
u(L)=0 :(X k„ e^‘' % A„ + PzX (-k„) '‘-"f') = 0 (22)
n=] list
All the equations can be put into the following matrix form
A.'
A:
A,
• = -{Terms containing Pz} (23)
A,
A.
A.
A.
1336
Required An values are numerically found from equations (18) or (23). Their
substitution to the appropriate forms of equations (10) and (11) yield the
required responses at any point on the beam.
3.RESULTS AND DISCUSSION
The theoretical model used in the study is shown in Figure 1 and has the
following geometric and material properties:
L=l(m), A =1.0*10"' (m\ h = 5.0*10-^(m), 15 = 4.17*10-* (m\
Cy = 15.625*10'* (m), J = 3.33* lO"' ' (m*), =7.26*10'* (m"'), p= 2700 (kg/m*),
r„ =2.85*10''* (m*), E = 7*10'“ (N/m*), G = 2.6* 10'“ (N/m*).
Structural damping for torsional vibrations is included through, complex
torsional rigidity as GJ*=GJ(l+z|3). For coupled vibrations, it is also included
through the complex flexural rigidity as El^(H-z'n).
First presented are the results for purely torsional vibrations. A bar assumed to
have the given L, p, lo, G and J values is considered. The bar is then restrained
at both ends by springs having the same torsional stiffness Ktor- A very low
damping, p=10'^, is assigned and the resonance frequencies are precisely
determined. It is found that, the introduction of a small Ktor introduces a very
low valued resonance frequency. That fundamental frequency increases with
increasing Ktor and as torsional constraint reaches to very high values, it
approaches to the fundamental natural frequency of torsionally fixed-fixed
beam. Table 1 gives the fundamental frequencies for a range of Ktor values and
Figure 2 represents the fundamental mode shapes for selected Ktor values.
Table 1: Uncoupled Fundamental Torsional Resonance Frequencies
((3=10’^,No Warping Constraint)
Ktor [N1
Frequency [Hz]
0
0.
10'^
1.606
10''
5.035
io“
14.683
10'
28.407
10^
32.678
10^
33.187
10^
33.239
;10*“
33.245
1337
Figure 6 on the other hand represents the low frequency torsional receptances
of the case in which the warping constraint is taken into consideration and
the ends are free to warp. This graph is included in order to show the variation
of fundamental torsional resonance frequencies for a range of Ktor values.
_ - K,3r=
K„,=
kK,„=
5*10':
1*10':
5-10"
1-10'
5-10'
1*10®
Figure 6. Fundamental Frequencies of Purely Torsional Vibrations
((3=0.01, x=0.13579[m], Warping Constraint Included, Ends are Free to Warp)
The second part of the study investigates the characteristics of doubly-coupled
vibrations. Now, the effects of each constraint are separately considered. A
channel having the given parameters is supported at its ends by springs Kt, Kr,
and Ktor- Warping constraint is included in the analysis and the ends are
assumed to warp freely. First analyzed is the effects of Ktor- For this Kt=10^°
[N/m] and Kr =10^° [N] are assigned at both ends of the channel and kept fixed
throughout the study. Ktor is varied and the frequencies are shown in Table 2.
Table 2. Effects of Ktnr in
Doublv-counled Vibrations
(Kt=10^° [N/m] and Kr=10^® [N], Warping Constraint Included)
A: First Torsion Dominated Frequency [Hz]
B: First Flexure Dominated Frequency [Hz]
Ktor [Nl
A
B
10-^
1.607
134.603
10''
5.069
134.936
10°
15.620
138.258
10'
39.725
169.247
10^
56.691
318.423
10^
59.528
422.061
10^
59.828
430.642
1338
0.0 0.2 0.4 0.6 0.8 1.0
ND LENGTH
Figure 2. Fundamental Mode Shapes of Purely Torsional Vibrations
((3=0, No Warping Constraint)
Then, the warping constraint To is included in the analysis and the results of
purely torsional vibrations are presented again. The beam had the same Ktor
values at both ends and the numerical values of the relevant parameters are
taken to be those previously defined. Figure 3 represents the fundamental
mode shapes for which the ends are free to warp, whereas Figure 4 shows the
mode shapes of the case in which there is no warping at the ends.
Figure 3. Fundamental Mode Shapes of Purely Torsional Vibrations
(P=0, Warping Constraint Included, Ends are Free to Warp)
(P=0, Warping Constraint Included, Ends Can Not Warp)
Figure 5 is drawn to highlight the effects of end warping. Both ends of the
channel are restrained with Ktor=l * lO’ [N] and all the other parameters of the
study are kept fixed. Figure 5 represents the direct torsional receptances of
two cases in which the ends of the channel are allowed to warp and not to
warp in turn. It can be seen that the prevention of end warping increases the
resonant frequencies.
Figure 5. Frequency Response of Purely Torsional Vibrations
([3=0.01, X =0.13579 [m], Ktor=l*10' [N]» Warping Constraint Included)
1340
It can be seen that, when it has lower values Ktor is more effective on the
torsion dominated resonance frequencies. For the higher Ktor values, the
effects are more apparent on the flexure dominated frequencies.
Figure 7 represents the direct flexural receptance of the channel for
a set of selected end stiffnesses. Torsion dominated resonances at 59.528 [Hz],
206.071 [Hz] and 476.649 [Hz] appear as spikes. The flexure dominated
resonance occurs at 422.061 Hz.
Figure 7. Frequency Response of Doubly-coupled Vibrations
(ti=0.001, P=0.001, x=0.13579[m], Warping Constraint Included,
Ends are free to warp, Kr=1.10^°[N], Kt=1.10^° [N/m], Ktor=l-10^ [N] )
Then the effects of the rotational spring, Kr, are considered. The ends of the
channel are assumed to be restrained with Kt=10^° [N/m] and Ktor =10^° [N].
The resulting frequencies are given in Table 3 for a range of Kr values.
Table 3. Effects of Kr in Doublv-coupled Vibrations
(Kt=10^° [N/m] and Ktor=10^° [N], Warping Constraint Included)
A; First Torsion Dominated Frequency [Hz]
B: First Flexure Dominated Frequency [Hz] _
Kr[N] _ A _ B
10*^
58.678
205.344
10°
58.679
205.358
10^
58.699
206.688
10^
58.858
218.055
10"
59.437
289.738
10^
59.799
405.667
10^
59.855
429.345
1341
It is seen that Kr is not effective on torsion dominated resonance frequencies,
but plays significant role for flexure dominated resonance frequencies.
Finally considered the effects of the transverse spring Kt. Again, the channel is
assumed to have very high Kr and Ktor values at both ends and Kt values are
varied. Table 4 shows the resonance frequencies.
Table 4. Effects of Kt in Doublv-coupled Vibrations
(Kr=10^° [N] and Ktor =10^° [N], Warping Constraint Included)
A: First Torsion Dominated Frequency [Hz]
B: First Flexure Dominated Frequency [Hz] _
Kt [N/m]
A
B
10'
1.369
70.455
10^
4.328
70.501
10^
13.592
70.973
10^
39.282
77.379
10=
58.095
159.665
10=
59.699
361.255
10*
59.860
431.215
It can be seen that the transverse stiffness, like torsional stiffness, effects both
flexure and torsion dominated frequencies.
4.CONCLUSIONS
In this study, a new analytical method is presented for the analysis of forced
vibrations of open section channels in which the flexible supports provide the
end constraints. The dynamic response of open section channels is a coupled
problem and their analysis requires the simultaneous consideration of all the
possible vibratory motions. The wave propagation approach is an efficient tool
for this complicated problem and the developed method is based on that.
The current method analyzes the forced, coupled vibrations of open section
channels. The channels, taken as Euler-Bernoulli beams, have uniform cross-
section and a single symmetry axis. That consecutively leads to the coupling
of flexural vibrations in one direction and torsional vibrations. The excitation
is assumed to be in the form of a harmonic point force, acting at the centroid.
1342
Various frequency response curves of uncoupled and coupled vibrations are
presented for a variety of different elastic end boundary conditions( which may
also have the inertial properties). The developed method, although aimed at
determining the forced vibration characteristics, is also capable of determining
the free vibration properties. This is also demonstrated by presenting various
mode shape graphs. It has been determined that the transverse and the torsional
stiffnesses play more significant role as compared to the rotational stifness.
The method can be used in analyzing the effects of multi point and/or
distributed loadings. This can simply be achieved by modifying the terms of
the forcing vector without increasing the order of the relevant matrix equation.
The developed method can also be used in the analysis of elastically
supported, triply-coupled vibrations of uniform channels. Results of that study
will be the subject of another paper.
REFERENCES
1. Gere, J.M. and Lin, Y.K., Coupled Vibrations of Thin-Walled Beams of
Open Cross-Section. J. Applied Mech Trans. ASME.,\9SZ, 80,373-8.
2. Lin, Y.K., Coupled Vibrations of Restrained Thin-Walled Beams.
J. Applied Mech. Trans. ASME., 1960, 82, 739-40.
3. Dokumaci, E., An Exact Solution for Coupled Bending and Torsional
Vibrations of Uniform Beams Having Single Cross-Sectional Symmetry.
JSoundandVib.Am, 119,443-9.
4. Bishop, R.E.D, Cannon, S.M. and Miao, S., On Coupled Bending and
Torsional Vibration of Uniform Beams. J.Sound and Fi'/)., 1989,131,457-64.
5. Cremer, L. and Heckl, y\..,Structure~ Borne Sound, Springer-Verlag,1988.
6. Mead, D.J. and Yaman, Y., The Harmonic Response of Uniform Beams on
Multiple Linear Supports: A Flexural Wave Analysis. J. Sound and Vib,
1990, 141,465-84
7. Yaman, Y. Wave Receptance Analysis of Vibrating Beams and Stiffened
Plates. PA Z). Ttew, University of Southampton, 1989.
8 Yaman, Y., Vibrations of Open-Section Channels: A Coupled Flexural and
Torsional Wave Analysis. (J. Sound and Vib, Accepted for publication)
1343
1344
THE RESPONSE OF TWO-DIMENSIONAL PERIODIC STRUCTURES
TO HARMONIC AND IMPULSIVE POINT LOADING
R.S. Langley
Department of Aeronautics and Astronautics
University of Southampton
Southampton S017 IBJ
ABSTRACT
Much previous work has appeared on the response of a two-dimensional
periodic structure to distributed loading, such as that arising from a harmonic
pressure wave. In contrast the present work is concerned with the response
of a periodic structure to localised forcing, and specifically the response of the
system to both harmonic and impulsive point loading is considered by
employing the method of stationary phase. It is shown that the response can
display a complex spatial pattern which could potentially be exploited to
reduce the level of vibration transmitted to sensitive equipment.
1. INTRODUCTION
Many types of engineering structure are of a repetitive or periodic
construction, in the sense that the basic design consists of a structural unit
which is repeated in a regular pattern, at least over certain regions of the
structure. An orthogonally stiffened plate or shell forms one example of an
ideal two-dimensional periodic structure in which the fundamental structural
unit is an edge stiffened panel. Although a completely periodic structure is
unlikely to occur in practice, much can be ascertained regarding the structural
dynamic properties of a real structure by considering the behaviour of a
suitable periodic idealization. For this reason, much previous work has been
performed on the dynamic behaviour of two-dimensional periodic structures,
with particular emphasis on free vibration and the response to pressure wave
excitation [1,2]. However, no results have yet been appeared regarding the
response of two-dimensional periodic structures to point loading (as might
arise from equipment mounts), and this topic forms the subject of the present
work. A general method of computing the response to both harmonic and
impulsive loading is presented, and this is then applied to an example system.
Initially the response of a two-dimensional periodic structure to harmonic
point loading is considered, and it is shown that the far-field response can be
expressed very simply in terms of the "phase constant" surfaces which
describe the propagation of plane waves. It is further shown that for
1345
excitation within a pass band two distinct forms of response can occur; in the
first case the amplitude of the response has a fairly smooth spatial distribution,
whereas in the second case a very uneven distribution is obtained and "shadow
zones" of very low response are obtained. The second form of behaviour is
related to the occurrence of caustics (defined in section 3.3), and the
distinctive nature of the response suggests that a periodic structure might be
designed to act as a spatial filter to isolate sensitive equipment from an
excitation source.
Attention is then turned to the impulse response of a two-dimensional periodic
structure. It is again shown that the response can be expressed in terms of the
phase constant surfaces which describe the propagation of plane waves. The
application of the method of stationary phase to this problem has a number of
interesting features, the most notable being the fact that four or more
stationary points can arise. It is found that a surface plot of the maximum
response amplitude against spatial position reveals features which resemble the
"caustic" distributions obtained under harmonic loading.
2. RESPONSE TO A HARMONIC POINT LOAD
2.1 Modal Formulation and Extension to the Infinite System
A two-dimensional periodic structure consists of a basic unit which is repeated
in two directions to form a regular pattern, as shown schematically in Figure
1. Each unit shown in this figure might represent for example an edge
stiffened curved panel in an aircraft fuselage structure, a three-dimensional
beam assembly in a roof truss structure, or a pair of strings in the form of a
"-f" in a cable net structure. The displacement w of the system can be
written in the form w{n,x), where n={n^ nf) identifies a particular unit and
x = {Xi X2 X3) identifies a particular point within the unit. The coordinate
system x is taken to be local to each unit, and the precise dimension of both
X and the response vector w will depend on the details of the system under
consideration.
The present section is concerned with the response of a two-
dimensional periodic structure to harmonic point loading of frequency co. In
the case of a system of finite dimension, the response at location (n,x) to a
harmonic force F applied at (0,jCo) can be expressed in the standard form [3]
»’(n,*)=EE
P 9
(1)
where rj is the loss factor, 4>p^(n,x) are the modes of vibration of the system
and (j}pg are the associated natural frequencies. The modes ^p^ which appear
in equation (1) are scaled to unit generalized mass, so that
1346
/!, «! V
(2)
where V represents the volume (or equivalent) of a unit and p(jc) is the mass
density. The present concern is with the response of an infinite system, or
equivalently the response of a large finite system in which the vibration decays
to a negligible level before reaching the system boundaries. In this case the
response is independent of the system boundary conditions, and it follows that
any analytically convenient set of modes can be employed in equation (1). As
explained in reference [4], it is expedient to consider the Born- Von Karman
(or "periodic”) boundary conditions, as in this case the modes of vibration can
be expressed very simply in terms of propagating plane wave components.
In this regard it can be noted from periodic structure theory [5] that a
propagating plane wave of frequency w has the general form
w(rt,jc)=i?^{^(jc)exp(zej«, +ie^n^nu)t)} , (3)
where and eo are known as the propagation constants of the wave (with -
T<e,<T and -7r<G2<T for uniqueness), and g{x) is a complex amplitude
function. By considering the dynamics of a single unit of the system and
applying Bloch’s Theorem [5], it is possible to derive a dispersion equation
which must be satisfied by the triad (w, 61,62) - by specifying Gj and €2 this
equation can be solved to yield the admissible propagation frequencies w. By
way of example, solutions yielded by this procedure for a plate which rests on
a grillage of simple supports are shown in Figure 2 (after reference [6]). It
is clear that the solutions form surfaces above the 61-62 plane - these surfaces
are usually referred to as "phase constant" surfaces, and a single surface will
be represented here by the equation a;= 0(61,62). The phase constant surfaces
always have cyclic symmetry of order two, so that 0(ei,62)=0(-6i,-62); for an
orthotropic system the surfaces also have cyclic symmetry of order four, and
therefore only the first quadrant of the 61-62 plane need be considered
explicitly, as in Figure 2.
The key point about the Born- Von Karman boundary conditions is that
a single propagating wave can fully satisfy these conditions providing and
6o are chosen appropriately. The conditions state that the left hand edge of the
system is contiguous with the right hand edge, and similarly the top edge is
contiguous with the bottom edge, so that the system behaves as if it were
topologically equivalent to a torus. If the system is comprised of XN2
units, then a propagating wave will satisfy these conditions if and
62^2 =2x^ for any integers p and q. Following equation (3), the displacement
associated with such a wave can be written in the form
1347
(4)
where ei^ and €2, are the appropriate values of the phase constants, and
<^^^=0(ej^,e2g)- Now since it follows that a wave of
frequency travelling in the opposite direction to will also satisfy the
boundary conditions. This wave say) will have the form
where it has been noted from periodic structure theory that reversing the
direction of a wave leads to the conjugate of the complex amplitude function
^(x). The two waves represented by equations (4) and (5) can be combined
with the appropriate phase to produce two modes of vibration of the system
in the form
KSn,x)
•
Re
Im
(6)
By adopting this set of modes it can be shown [4] that equation (1) can be re¬
expressed as
K.(«,x)= £ £ 2g;WF^g„(xJexp(-ie,^«,-%,n,)
^=1-^/2 9=1 -Nj/2 +Z?7) -(J?
where and M have been taken to be even, and the amplitude function gp^
is scaled so that
g^,=[2p(x)WV,Ar,]-%(x), {llV)\^f„(x)f;,(x)dx=\, (8,9)
where the normalized amplitude function fp^ is defined accordingly. The
summation which appears in equation (7) includes only those modes associated
with a single phase constant surface 0(€i,62); if more than one surface occurs
then the equation should be summed over the complete set of surfaces. The
summation will include modes for each surface, which is consistent with
known results for the modal density of a two-dimensional periodic structure.
Equation (7) yields the response of a finite system of dimension XM
to a harmonic point load - this response is identical to that of an infinite
system if the vibration decays to a negligible amount before meeting the
system boundaries. If the system size is allowed to tend to infinity in equation
1348
(7) then neighbouring values of the phase constants e^p and €2^ become closely
spaced (since deip=ei_p+i-eip=2T/A^i and de2g= €2,9+ 2x77/2) > and in this
case the summations can be replaced by integrals over the phase constants to
yield
- LI me„e,)ninv)-c^^
where 0)= 0(61,62) and ^(x) is the complex amplitude associated with the wave
(£0,61,62). The evaluation of the integrals which appear in equation (10) is
discussed in the following sub-sections.
2.2 Integration over 61
The integral over 6i which appears in equation (10) can be evaluated by using
contour integration techniques. Two possible contours in the complex e, plane
are shown in Figure 3; to ensure a zero contribution from the segment
Im(6i) = ±oo, the upper contour is appropriate for /Zi<0 while the lower
contour should be used for n^X). For each contour the contributions from
the segments and 61;;= x cancel, since the integrand which appears in
equation (10) is unchanged by an increment of 2x in the real part of ei. The
only non-zero contribution to the integral around either contour therefore
arises from the segment which lies along the real axis. The poles of the
integrand occur at the 61 solutions of the equation
[fl(6i,62)?(U/i7)-a;^=0, (11)
for specified 62 and oj. By definition there will be two real solutions^ in the
absence of damping (77=0) providing the frequency range covered by the
phase constant surface includes oj. Any complex solutions to equation (11) in
the absence of damping will correspond physically to "evanescent" waves
which decay rapidly away from the applied load. The present analysis is
concerned primarily with the response of the system in the far field (that is,
at points remote from the excitation source), and for this reason attention is
focused solely on those roots to equation (11) which are real when 77=0. The
effect of damping on these roots can readily be deduced: if 77 is small then it
follows from equation (11) that a real solution 6^ will be modified to become
6I-i(o7/2)(5Q/^6l)■^ and hence the real pole for which dn/36i<0 is moved to
the upper half plane, while that for which 30/56, >0 is moved to the lower
half plane. Given that the residue at such a pole is proportional to (30/36i)'\
^One positive and one negative. These solutions will have the form ±6,
for an orthotropic system.
1349
it follows that the sign of the residue which arises from the contour integral
is determined by the integration path selected, and hence by the sign of
These considerations lead to the result
~Tr~l fi|3fi/3«,|(l+/)/)
(12)
where e,(£2,M) is the appropriate solution to equation (11). The evaluation of
the integral over €3 is discussed in the following section.
3.3 Integration over €3
Since the present concern is with the response of the system at some distance
from the excitation point, the integral over e, which appears in equation (12)
can be evaluated to an acceptable degree of accuracy by using the method of
steepest descent [7], With this approach it is first necessary to identify the
value of €2 for which the exponent -i(eirti+e2«2) is stationary. The condition
for this is
{3ejde^n^+n^=0. (13)
Now Gi and satisfy the dispersion relation, equation (11), and thus equation
(13) can be re-expressed in the form
(aQ/a62)«r(5^2/a€,)«2=o, (i4)
where it has been noted from equation (11) that, for fixed co, 3ei/3€2=-
(5Q/3e2)/(9fi/36i), In the absence of damping the wave group velocity lies in
the direction (SQ/Sei and in this case it follows from equation (14)
that the group velocity associated with the required value of €2 is along («i ru).
For light damping this result will be substantially unaltered, although damping
will have an important effect on the value of the exponent -\{€^n^-\-e2iv^ at the
stationary point. This effect can be investigated by noting initially that
d{e^n^+&^n^)IBr}~{deJbri B&Jbr]).{n^ n^. (15)
Now it follows from equation (11) that for light damping {ri<l)
(dQ/de, dQlde,).{de,ldr] de^/dv) = -io)l2, (16)
and hence equations (14)-(16) can be combined to yield the following result
at the stationary point
d{e^n^+e^n2)/dr) = -io)n/2c^. (17)
1350
Here Cg=^{{d£l!dex?+{d^lbe^'^] is the resultant group velocity and
n=V[ni+ni-'\ is the radial distance (in units) from the excitation point to the
unit under consideration. It follows that in the immediate vicinity of the
stationary point the exponent can be expanded in the form
(18)
where the subscript 0 indicates that the term is to be evaluated at the
stationary point under the condition 7]=0; for ease of notation, this subscript
is omitted in the following analysis. The method of steepest descent proceeds
by substituting equation (18) into equation (12) and assuming that: (i) the main
contribution to the integral arises from values of 62 in the immediate vicinity
of the stationary point; (ii) the integrand is effectively constant in this vicinity,
other than through variation of the term e2-(e2)o which appears in equation
(18); (iii) under conditions (i) and (ii) the integration range can be extended
to an infinite path without significantly altering the result. The method then
yields [7]
w{n,x) = ~if *F7o[20V|aQ/a£j/2xp(A:)p(A:o)|«i(aV5e2)| .
(19)
where/ is the normalized complex wave amplitude which appears in equation
(9), and all terms are to be evaluated at the stationary point.
The stationary point associated with equation (19) is that point for
which the group velocity is in the {n^ rQ direction. Geometrically, this is the
point at which the normal to the curve a)=Q(ei, £2) itt the plane lies in the
{n^ Ho) direction. Three such curves are shown schematically in Figure 4,
together with a specified (n, 722) direction. For the frequencies and coj the
situation is straight forward, in the sense that a unique stationary point exists
for any {n^ n^) direction. For the frequency 0J2 the situation is more complex,
since: (i) two stationary points occur for the (n^ rQ direction shown, and (ii)
no stationary point exists if the {n^ direction lies beyond the heading B
shown in the figure (the dashed arrow represents the normal with maximum
inclination to the axis). In case (i) equation (19) should be summed over
the two stationary points, while in case (ii) the method of steepest descent
predicts that w{n,x) will be approximately zero, leading to a region of very
low vibrational response. If the direction («i coincides with the dashed
arrow, then equation (19) breaks down, since it can be shown that
at this point. The heading indicated by the dashed arrow represents a caustic
[7], and the theory given in the present section must be modified for headings
1351
(Wi Wj) which are in the immediate vicinity of the caustic - full details of the
appropriate modifications are given in reference [4]. An example which
illustrates the application of equation (19) is given in section 4.
3. RESPONSE TO AN IMPULSIVE POINT LOAD
If the system is subjected to an impulsive (i.e. a delta function applied at
r=0), rather than harmonic, point load, then equation (10) becomes [8]
w{n,x,t)={N^NJ2Tp) f [ g*
II (20)
where co = ^](€i,€2)- The method of stationary phase can be applied to this
expression to yield [8]
w{n,x,t)-{\l2yr)[p{x)p{x)\J\r‘^^^^
exp(-/Gjtti -k^n^nQt+ib) } ,
(21)
where all terms are evaluated at the stationary point, and J and 5 are defined
as
/=(a"n/a6?)(a"Q/ae^-(3"0/ae,a6,)", S=(Tr/4)sgn(a%/fle?){l+sgn(/)}.
(22,23)
In this case the stationary point is given by the solution to the equations
=(afi =(30 /3e,>. (24,25)
In practice equations (24) and (25) may yield multiple solutions (stationary
points), in which case equation (21) should be summed over all such points.
Furthermore, stationary points having 7=0 indicate the occurrence of a
caustic, and equation (21) must be modified in the immediate vicinity of such
points as detailed in reference [8]. An example of the application of equation
(21) is given in the following section.
1352
4. EXAMPLE APPLICATION
4. 1 The System Considered
The foregoing analysis is applied in this section to a two-dimensional periodic
structure which consists of a rectangular grid of lumped masses m which are
coupled through horizontal and vertical shear springs of stiffness and h
respectively. Each mass has a single degree of freedom consisting of the out-
of-plane displacement w, and a linear spring of stiffness k is attached between
each mass and a fixed base. It is readily shown that the system has a single
phase constant surface of the form
Q -(ep€,)=iLii(l-cos€i)+/x2(l-cos€2)+a;^,
where ix^^lkjm, and o)^-==klm. The function U can be used in
conjunction with the analysis of the previous sections to yield the response of
the system to harmonic and impulsive point loading; in this regard it can be
noted that for the present case p{x)=m, V—l, and /(:»:) = 1.
4.2 Response to Harmonic Loading
The surface is shown as a contour plot in Figure 5 for the case
m=1.0, oj„==0, Ati = 1.0, Results for the forced harmonic response
of this system at the two frequencies w = 1.003 and cx> = 1.181 are shown in
Figures 6 and 7. In each case the response of a 40 x40 array of point masses
is shown; a unit harmonic point load is applied to mass (21,21) and the loss
factor is taken to be =0.05. Two sets of contours are shown in each Figure:
the smooth contours have been calculated by using equation (19) while the
more irregular contours have been obtained by a direct solution of the
equations of motion of the finite 1600 degree-of-freedom system. By
considering the results shown in Figure 6, it can be concluded that: (i) for the
present level of damping the finite system effectively behaves like an infinite
system, and (ii) the analytical result yielded by equation (19) provides a very
good quantitative estimate of the far field response. It can be noted from
Figure 5 that no caustic occurs for a) = 1.003, in the sense that equation (14)
yields only one stationary point which contributes to equation (19). In
contrast, a caustic does occur for the case a) = 1.181, and this leads to the very
irregular spatial distribution of response shown in Figure 7. Two stationary
points contribute to equation (19), and constructive and destructive
interference between these contributions is responsible for the rapid
fluctuations in the response amplitude. It is clear that the response exhibits
a "dead zone" for points which lie beyond the caustic heading (in this case
30.25° to the «i-axis), as predicted by the analysis presented in section 2.
4.3 Response to Impulsive Loading
The impulse response of a system having m=1.0, /xi = 1.0, ^2=0. 51, and
1353
co„'“0.25 has been computed. The impulse was taken to act at the location
/Zi=«2=0 and the time history of the motion of each mass in the region -
iO<(72i,n2) ^ 10 was found by using equations (21)-(23). For each mass the
maximum response \w\ was recorded, and the results obtained are shown as
a contour plot in Figure 8. In accordance with Fourier’s Theorem, the
impulse response of the system contains contributions from all frequencies,
and therefore the spatial distribution of | w| can be expected to lie somewhere
between the two extreme forms of harmonic response exhibited in Figures 6
and 7. This is in fact the case, and the response shown in Figure 8 retains a
distinctive spatial pattern. As discussed in reference [8], the results shown in
Figure 8 are in good agreement with direct simulation of the impulse response
of the system.
5. CONCLUSIONS
This paper has considered the response of a two-dimensional periodic structure
to both harmonic and impulsive point loading. With regard to harmonic
loading, it has been shown that the spatial pattern of the response is strongly
dependent on the occurrence of a caustic: if no caustic occurs then the
response has a fairly smooth spatial distribution, whereas the presence of a
caustic leads to an irregular spatial distribution and a "dead zone" of very low
response. This type of feature is also exhibited, although to a lesser degree,
in the spatial distribution of the response to an impulsive point load. This
behaviour could possibly be exploited to reduce vibration transmission along
a specified path, although the practicality of this approach for a complex
system has yet to be investigated. The present analytical approach can be
applied to all types of two-dimensional periodic structure - the information
required consists of the phase constant surface(s) 0(61,62) and the associated
wave form(s) f{x), both of which are yielded by standard techniques for the
analysis of free wave motion in periodic structures [1,2].
REFERENCES
1. S.S. MESTER and H. BENAROYA 1995 Shock and Vibration 2, 69-
95. Periodic and near-periodic structures.
2. D.J. MEAD 1996 Journal of Sound and Vibration 190, 495-524.
Wave propagation in continuous periodic structures: research
contributions from Southampton 1964-1995.
3. L. MEIROVITCH 1986 Elements of Vibration Analysis, Second
Edition. New York: McGraw-Hill Book Company.
4. R.S. LANGLEY 1996 Journal of Sound and Vibration (to appear).
The response of two-dimensional periodic structures to point harmonic
forcing.
1354
1
■
■
■
1
■
■
■
0
m
■
1
■
Xj. ■j'
Oi
< -
^1
1 ^
Figure I , Schematic of a two-dimensional periodic structure. The arrow indicates' the reference unit
(with n=0) while the circle represents a general point (re,x). The structure may have a third spatial
coordinate X3, which for convenience is not shown in the figure. The point load considered in section
3 is applied at the location of the arrow.
Figure 2. Phase constant surfaces for a plate which rests on a square grillage of simple supports.
Q is a non-dimensional frequency which is def as Q=a)LV(m/D), where m and D are respectively
the mass per unit area and the flexural rigidii^ the plate, and L is the support spacing.
1355
Figure 5. Contour plot of the phase constant surface for the case ^=0.57. The contours
are separated by an increment Aco =0. 1477. The two contours considered in section 4.2 are indicated
as follows: (a) a) = 1.033; (b) a) = 1.181.
Figure 6. Response |H'(n,x)|2 of the 40x4v .uass/spring system to a unit harmonic force of
frequency w- 1.033 applied at the location i=y=21. The contours correspond to the response levels
lK«,x)!'=0.01, 0.02, andO.05.
1357
STICK-SLIP MOTION OF AN ELASTIC SLIDER SYSTEM ON
A VIBRATING DISC
HOuyang J E Mottershead M P CartmeU ' MIFiiswell^
Department of Mechanical Engineering, University of Liverpool
^ Department of Mechanical Engineering, University of Edinburgh
^ Department of Mechanical Engineering, University of Wales Swansea
ABSTRACT
The in-plane vibration of a slider-mass which is driven around the surface of a
flexible disc, and the transverse vibration of the disc, are investigated. The disc
is taken to be an elastic annular plate and the slider has flexibility in the
circumferential (in-plane) and transverse directions. The static fiiction
coefi&cient is assumed to be higher than the kinetic friction. As a result of the
fiiction force acting between the disc and the slider system, the slider will
oscillate in the stick-slip mode in the plane of the disc. The transverse vibration
induced by the slider will change the normal force of the slider system acting
on the disc, which in turn will change the in-plane oscillation of the slider. For
different values of system parameters, the coupled in-plane oscillation of the
slider and transverse vibration of the disc will exhibit quasi-periodic as well as
chaotic behaviour. Rich patterns of chaotic vibration of the slider system are
presented in graphs to illustrate the special behaviour of this non-smooth
nonlinear dynamical system The motivation of this work is to analyse and
understand the instability and/or squeal of physical systems such as car brake
discs where there are vibrations induced by non-smooth dry-fiiction forces.
NOMENCLATURE
a , b mner and outer radii of the annular disc
c damping coefficient of the slider in in-plane direction
h thickness of the disc
i = V=T
1359
k , k transverse and in-plane stiffiiess of the slider system
m
r
t
^stiek
mass of the slider
radial co-ordinate in cylindrical co-ordinate system
radial position of the slider
modal co-ordinate for k nodal circles and / nodal diameters
time
the time of the onset of sticking
u , Mp transverse displacement of the slider mass and its initial value
w , Wq transverse displacement of the disc and its initial value
flexural rigidity of the disc
Kelvin-type damping coefficient
Young’s modulus
initial normal load on the disc jfrom the slider system
total normal force on the disc from the slider system
combination of Bessel functions representing mode shape in radial
direction
circumferential co-ordinate of cylindrical co-ordinate system
kinetic and static dry fiction coefficient between the shder and the
D
D'
E
N
P
R.
0
disc
V
P
(P
^ stick
¥
¥ki
CO.
Poisson’s ratio
damping ratio of the disc
specific density of the disc
absolute circumferential position of the slider
absolute circumferential position of the slider when it sticks to the disc
circumferential position of the slider relative to the drive point
mode function for the transverse vibration of the disc corresponding to
^kl
natural (circular) frequency correq)onding to <5^^
1360
Q constant rotating speed of the drive point around the disc in radians per
second
INTRODUCTION
There exists a whole class of mechanical systems which involve discs rotating
relative to stationery parts, such as car brake discs, clutches, saws, computer
discs and so on. In these systems, dry-friction induced vibration plays a crucial
role in system performance. If the vibration becomes excessive, the system
might fail, or cease to perform properly, or make offensive noises. In this
paper, we investigate the vibration of an m-plane slider system, with a
transverse mass- spring- damper, attached through an in-plane spring to a drive
point which rotates at constant speed around an elastic disc, and the vibrations
of the disc. Dry friction acts between the sHder system and the disc.
Dry-friction induced vibration has been studied extensively [1-4]. For car brake
vibration and squeal, see the review papers [5,6]. The stick-slip phenomenon of
dry-friction induced vibration is studied in the context of chaotic vibration [7-
10]. Popp and Stelter [7] studied such motion of one and two degrees of
freedom system and foimd chaos and bifiircation. They also conducted
experiments on a beam and a circular plate (infinite number of degrees of
freedom). These theoretical works are about systems of less than three degrees
of freedom, and the carrier which activates the friction is assumed to be rigid.
In this paper, we consider an elastic disc so that the transverse vibrations of the
disc are important. As a result of including the transverse vibrations of the disc,
rich patterns of chaos, which have not been reported previously are found. If
there is only shding present at constant speed, the problem is reduced to a
linear parametric analysis which was carried out for a pin-on- disc system in
[11] and for a pad-on-disc system in [12,13].
m-PLANE OSCILLATION OF THE SLIDER SYSTEM
As the drive point, which is connected to the shder-mass through an in-plane,
elastic spring, is rotated at constant angular speed around the disc, the driven
slider will undergo stick-slip oscillations. The whole system of the shder and
the disc is shown in Figure 1.
The equation of the in-plane motion of the slider system relative to the rotating
drive point, in the sliding phase, is.
1361
(1)
while in sticking, the equation of the motion becomes.
The relationship between the relative motion of the slider system to the drive
point and its absolute motion (relative to the stationary disc) is
(p = Qt + y/, (3)
We consider the foEowing initial conditions which are intended to simulate
what happens in a disc brake. The slider system is at rest and there is no normal
loading on the disc j&om the slider. Then a constant normal load is applied
which causes transverse vibrations in the disc. At the same time, the drive point
is given a constant angular velocity. Other initial conditions are possible, so
that there is no loss of generality.
First, sliding from the initial sticking phase occurs when,
The slider will stick to the disc agaiu when,
ju^P (during sliding), (5)
or it will begin to sHde again iJ^
ij/ = , \k^rQtp-\ > (dxiring sticking). (6)
Consequently, the slider system will stick and slide consecutively on the disc
surface.
TRANSVERSE VIBRATION OF THE ANNULAR DISC
The equation of motion of the disc under the slider system is,
ph^ + D'V^'w + DW = --5(r-r„)5i$-<p)P. (7)
a/" r
The total force P is the summation of initial normal load N and the resultant
of the transverse motion w of the slider. Its expression is,
P - N + mu + cu-h kiu-u^) . (8)
Since it is assumed that the slider system is always in perfect contact with the
disc, then,
1362
(9)
u{t) = w{r„(p{t)A-
Substitution of ecpiations (8) and (9) into (7) leads to.
ph
^ — + /:)VV = --5(r - r, )6(^ - (p)[N +
dt
dt
^..dw d^w d^w .dw dw
(10)
k(w-wj\.
Note that equation (10) is valid whether the slider system is sticking or sliding.
When the slider sticks to the disc, equation (10) reduces to.
+ D-V‘ ^ +£)W = --5(r - )5(5 -<p)x
St St r (11)
d'^W ^ M
[A^+m— + c— + A:(w-w„)].
COUPLED VIBRATIONS OF THE SLIDER AND THE DISC
Assume that the transverse motion of the disc can be represented by,
M>{r,e,t)=ttii/,{r,0)q,it), (12)
Jt=0 /=-<io
and,
where {r) is a combination of Bessel functions satisfying the boundary
conditions in radial direction at the inner radius and outer radius of the disc.
The modal functions satisfy the ortho-normality conditions ofj
(14)
1363
Equations (10) and (11) can be simplified by being written in terms of the
modal co-ordinates from equation (12).
During sticking, the motion of the whole system of the slider and the disc can
be represented by,
N
+ 2^0) -
— Z^„('-o)^«('-o)exp[i(5-0(»]x (15)
ph.U ''=0 s=-«o
{mq„+cq^+k{q^-q„{^)\).
The sticking phase can be maintained i^
rA\¥\</^XJ^+-
^Z Z ^«(?')exp(i/?»)x
{mq„+cq„+ k[q„ - (0)] } ] .
While in sliding, the motion of the whole system can be represented by.
%+2^a>^,qu+a>lq„ =
^^«('i)exp(-i/<»)-
-rrri Z^„('-o)^«(n,)exp[i(5-0«!’]x
pnO r=<i 5^=-oo
H?,. +i2sw„ +isw„) +
k{q„-qS'^)]},
r„{m{j/ + kw) = -/i,Sign((Z))[A'' +-
^Z Z KMx
exp(LS(Z>)M?„ +i2s^„ +(ls^-5>^)9„] +
c{q„ +>s(?g„) +*[!?„ -?„(o)]}].
The sliding phase can be maintained if,
1364
\K^M< mXN +-J^t t K{r,)es.p{is(p)x
■yjphb ^=0^^
{iTiq„+i2sj>q„+(is^-s^j)^)qJ+ (20)
c(?„ +isj^J + k[q„ -9„(0)]}],
when,
\j/ = ~0 or ^ = 0. (21)
COMPUTING PROCESS
As the shder system sticks and sHdes consecutively, the governing equations of
the coupled motions of the whole system switch repeatedly from equations
(15), (16) and (17) to equation (18), (19) and (20). TTie system is not smooth.
Since the condition which controls the phases of the slider system itself
depends on the motions, it is also a nonlinear system, whether is a constant
or a function of relative speed ^ . In order to get modal co-ordinates, we have
to truncate the mfinite series in equation (12) to jfinite terms. Then numerical
integration is used to solve equations (15), (18) and (19). Here a fourth order
Rimge-Kutta method is used for second order simultaneous ordinary
differential equations.
Since equation (18) has time-dependent coefficients, time step length has to be
very small. Constant time step lengths are chosen when the m-plane slider
motion is well within the sticking phase or the shding phase in the numerical
integration. As it is imperative that the time step should be chosen such that at
the end of some time intervals the shder happens to be on the sticking- shding
interfaces, we use a prediction criterion to choose next time step length when
approaching these interfaces. Therefore, at the sticking-shding interfaces, the
time step length is variable (actuahy smaher than it is while weh within sticking
or shding). Nevertheless, tbe interfaces equations ( equation (17) or equations
(20-21) ) are only approximately satisfied [10].
When transverse motion of the shder system becomes so violent that the total
normal force P in it becomes negative or becomes several times larger than
the initial normal load 7/ , we describe the system as being unstable. Then the
motion begins to diverge. But this instabihty should be distinguished from a
chaotic motion which is bounded but never converge to a point.
1365
NUMERICAL EXAMPLES
The following data are used in the computation of numerical examples:
a = 0.065m, b = 012m, OTm, /z = 0.001m; = 120GPa, v= 0.35,
Z)*= 0.00004; yW, =0.4, //^=0.24, A: = lOOON/m, = lOON/m,
m = 0.1kg, p = 7000kg / m^ The disc is clamped at inner radius and free at
outer radius. Note that in these numerical examples, the disc thickness is
dehherately taken to be very small in order to reduce the amount of computing
work. However, this will not affect the qualitative features of the results or
conclusion drawn from the results thus obtained. The first five natural
(circular) frequencies are 451.29, 462.73, 426.73, 508.23, 508.23. We will
concentrate on the vibration solutions at different levels of initial normal load.
But occasionally solutions at different rotating speed or different damping
ratios are investigated. Unless specified expressly, the Poincare sections are for
the in-plane vibration of the slider system
First of all, we study the effect of the normal loadA^. Take f2=l0 and
^ — When N is very small^ the Poincare section is a perfect ellipse
which indicate the in-plane vibration of the slider system is quasi-periodic, as
the transverse vibration of the disc is too small to affect total normal force P .
A typical plot of such motion is shown in Figure 2 for A^=0.5kPa. As N
increases, the sticking period gets longer, the bottom part of the elhpse evolves
into a straight line, indicating phase points within the sticking phase. One of
such plots is given in Figure 3 for N =3kPa. A further increase of N not only
lengthens the straight line part of the Poincare section, but also creates an
increasingly ragged outline in the arch part of the plot. The curve is no longer
smooth and it seems that the in-plane motion begins to enter a chaotic state
from the quasi-periodic state. Figure 4 presents the Poincare section plot for
N =7.5kPa. There is a transition period from quasi-periodic motion to chaotic
motion, extending from N =6kPa up to N =9kPa. Chaos becomes detectable
at iV=10kPa, whose Poincare section is shown in Figure 5. Then chaotic
vibration follows. When N ~15kPa, the arch part of the Poincare section
becomes so fuzzy and thick that it should no longer be considered as a curve,
but rather a narrow (fractal) area. A hlow-up’ view of the arch part reveals
that phase points are distributed across the arch. Both plots are ^own in
Figure 6. Between iV'=17.ikPa and 18.325kPa, the vibration of the slider
enters a new stage, with Poincare sections looking like star clusters as
illustrated in Figures 7 and 8. This kind of motions are rather extraordinary and
1366
have not been reported in other works on stick-shp motions with a rigid
carrier. Afterwards, the ‘arch-door’ hke Poincare sections come back (see
Figure 9). The difference from previous Poincare sections of lower N is that
the new Poincare sections look like overlapping of earlier Poincare sections,
which indicates a clear layered structure, as diown in Figure 10 and more
obviously in the left hand side of Figure 11. At this stage, the vibration is very
chaotic. To give the reader a better picture, the Poincare section of a fixed
point on the disc at ( = 0.1m and = 0 ), is also shown in the right hand side
of Figure 11. The Poincare sections of the slider-mass and a point on the disc
are also given in Figures 12-15. In Figure 12 for A^=30.5kPa, the vibration
goes unstable. Here again, the Poincare sections have not been reported
elsewhere.
If disc damping is increased, vibration will become more regular, as shown in
Figure 13. Comparing Figures 11 with 13, we see that increase of disc damping
makes the vibration more concentrated though not always smaller. Unstable
vibration can be stabilised with more disc damping, as seen from Figure 14.
If there is no damping at aU, the resulting vibration due to dry fiiction will be
unstable, even at very small normal load N . In Figure 15, the motion of the
slider tends to run away in the tangential direction from the normal ellipse
attractor, while the motion of the disc goes unbounded.
Increasing the speed of the drive point seems to make vibration more chaotic
and more unstable, as shown in Figures 16-18. At this stage, however, we are
unable to make a definite conclusion on rotating speed as there might be
intervals of regular motions and intervals of chaotic motion for . More
numerical examples must be computed to draw a positive conclusion on this
parameter.
The correlation dimension is not a good measure of the vibration for the
current problem because its values fluctuate in some numerical examples. This
failure was perhaps first discovered in [7]. The reason can be either that the
system is non-smooth, or that the system has multiple degrees of freedom, or
both. Therefore, the correlation dimension or any other fractal dimensions is
not presented in this paper.
CONCLUSIONS
In this paper, we studied the in-plane stick-slip vibration of a slider system with
a transverse mass-spring-damper driven around an elastic disc through a spring
from a constant speed drive point, and transverse vibrations of the disc. The
1367
whole system had been reduced to six degrees of freedom after simplification.
From numerical examples computed so far, we can conclude that:
1. Both vibrations are very complex as this is a multi-degree of freedom, non¬
smooth system Rich patterns of chaotic vibration are found. Some have not
been reported elsewhere.
2. For the normal pressure parameter, smaller values allow quasi-peiiodic
solutions. Greater pressures result in chaotic motions. At certain large
pressures, the vibrations become unstable.
3. Disc dartping makes vibration more concentrated to smaller areas and when
sufficiently large it can stabilise otherwise unstable vibration.
4. An increase in the rotating speed can make the vibration more chaotic or
more unstable.
5. Correlation dimension is not a good measure of the vibration of this multi¬
degree of freedom, non-smooth dynamical system
6. Much more investigation needs to be earned out in understanding and
characterising the vibration of multi- degree of freedom, non- smooth dynamical
systems.
ACKNOWLEDGEMENT
This research is supported by the Engineering and Physical Sciences Research
Council (grant niunber J35177) and BBA Friction Ltd.
REFERENCES
1. Nakai, M., Chiba, Y. and Yokoi, M., Railway wheel squeal. Bulletin of
JSME, 1984, 27, 301-8
2. Lin, Y-Q and Wang Y-H, Stick-sHp vibration of drill strings. XEng.Ind.,
TransASME, 1991, 113, 38-43
3. Ferri, A. A. and Bindemann, A.C., Damping and vibration of beams with
various types of fiictional support conditions. J. VibAcoust, TransASME^
1992, 114, 289-96
4. Lee, A.C., Study of disc brake noise using multi-body mechanism with
friction interface. In Friction-Induced Vibration, Chatter, Squeal, and
Chaos, Ed. Ibrahim, KA. and Soom, A., DE-Vol. 49, ASME 1992, pp.99-
105
1368
5. Ibrahim, R.A., Friction-induced vibration, chatter, squeal, and chaos. In
Friction-Induced Vibration, Chatter, Squeal, and Chaos, Ed. Ibrahim,
KA. and Soom, A, DE-VoL 49, ASME 1992, pp. 107-38
6. Yang, S. and Gibson, R.F., Brake vibration and noise: reviews ,comments,
and proposed considerations. Proceedings of the 14th Modal Analysis
Conference, The Society of Experimental Mechanics, Inc., 1996, pp.l342-
9
7. Popp, K and Stelter, P., Stick-slip vibration and chaos. Phil. Trans. R Soc.
Lond. A(1990), 332, 89-106
8. Pfeiffer, F. and Majek, M., Stick-slip motion of turbine blade dampers. Phil.
Trans. R Soc. Lond. A(1992), 338, 503-18
9. Wojewoda, J., Kapitaniak, T., Barron, R. and Brindley, J., Complex
behaviour of a quasiperiodically forced experimental system with dry
friction. Chaos, Solitons and Fractals, 1993, 3, 35-46
10. Wiercigroch, M., A note on the switch function for the stick-slip
phenomenon. J.SoundVib., 1994, 175, 700-4
11. Chan, S.N., Mottershead, J.E. and Cartmell, M.P., Parametric resonances
at subcritical speeds in discs with rotating frictional loads. Proc. Instn.
Mech. Engrs, 1994, 208, 417-25
12. Mottershead, J.E., Ouyang, R, Cartmell, M.P. and Friswell, M.I.,
Parametric Resonances in an annular disc, with a rotating system of
distributed mass and elasticity; and the effects of friction and dan:q)ing.
Proc. Royal Soc. Lond. A., 1997, 453, 1-19
13. Ouyang, H., Mottershead, J.E., Friswell, M.l. and Cartmell, M.P., On the
prediction of squeal in automotive brakes. Proceedings of the 14th Modal
Analysis Conference, The Society of Experimental Mechanics, Inc., 1996,
pp. 1009-16
Figure 1. Slider system and disc in cylindrical co-ordinate system
1369
Figure 9. iV=21kPa
Figure 10. iV=24kPa
1374
A Finite Element Time Domain
Multi-Mode Method For Large Amplitude
Free Vibration of Composite Plates
Raymond Y. Y. Lee, Yucheng Shi and Chuh Mei
Department of Aerospace Engineering
Old Dominion University, Norfolk, VA 23529-0247
Abstract
This paper presents a time-domain modal formulation using the finite element method for
large-amplitude firee vibrations of generally laminated thin composite rectangular plates. Accurate
fi'equency ratios for fundamental as well as higher modes of composite plates at various maximum
deflections can be determined. The selection of the proper initial conditions for periodic plate
motions is presented. Isotropic beam and plate can be treated as special cases of the composite
plate. Percentage of participation from each linear mode to the total plate deflection can be
obtained, and thus an accurate frequency ratio using a minimum number of linear modes can be
assured. Another advantage of the present finite element method is that the procedure for obtaining
the modal equations of the general DujB5ng-type is simple when compared with the classical
continuum Galerkin’s approach. Accurate frequency ratios for isotropic beams and plates, and
composite plates at various amplitudes are presented.
Introduction
Large amplitude vibrations of beams and plates have interested many
investigators [1] ever since the first approximate solutions for simply supported
beams by Woinowsky-Kiieger [2] and for rectangular plates by Chu and Herrmann
[3] were presented. Singh et al. [4] gave an excellent review of various formulation
and assumptions , including the finite element method for large amplitude firee
vibration of beams. Srirangaraja [5] recently presented two alternative solutions,
based on the method of multiple scales (MMS) and the ultraspherical polynomial
approximation (UP A) method, for the large amplitude firee vibration of a simply
supported beam. The fi'equency ratios for the fundamental mode, ca/C0L> at the ratio
of maximum beam deflection to radius of gyration of 5.0 (Wmax/r =5.0) are 3.3438
and 3.0914, using the MMS and the UPA method, respectively. Eleven firequency
ratios including nine firom reference [4] were also given (see Table 1 of reference
[5]). It is rather surprising that the firequency ratio for the fundamental mode at
Wn«x/r =5.0 for a simply supported beam varied in a such wide range: fi-om the
lowest of 2.0310 to the highest of 3.3438, and with the elliptic function solution by
Woinowsky-Kiieger [2] giving 2.3501. Similar wide spread exists for the vibration
of plates. Rao et al [6] presented a finite element method for the large amplitude
firee flexural vibration of unstiffened plates. For the simply supported square plate.
1375
frequency ratios from six different approaches were reported (see Table 1 of
reference [6]). The frequency ratio at Wmax/h =1.0 varied from a low of 1.2967 to a
high of 1.5314, with Chu and Herrmann’s analytical solution [3] at 1.4023.
This paper presents a finite element time domain modal formulation for the large
amplitude free vibration of composite plates. The formulation is an extension from
the isotropic plates [7], and the determination of initial conditions for periodic
motions was not employed in reference [7]. The convergence of the fundamental
frequency ratio is investigated for a simply supported beam and a simply supported
square plate with a varying number of finite elements and a varying number of
linear modes. Accurate frequency ratios for fundamental and higher modes at
various maximum deflections, and percentages of participation from various linear
modes, are obtained for beams and composite plates.
Formulation
Strain-Displacement and Constitutive Relations
The von Karman strain-displacement relations are applied. The strains at any
point z through the thickness are the sum of membrane and change of curvature
strain components:
{e} =
-w.xx
V,y
>+<
/ 2
> + z<
-W,yy
U,y+V,x^
W,yj
= {Em} + {sb} + z{K}
(1)
where and {eb}are the membrane strain components due to in-plane
displacements u and v and the transverse deflection w, respectively. The stress
resultants, membrane force {N} and bending moment {M}, are related to the strain
components as follows:
■[A] IB]'
[B1 [Di;
(2)
where [A] is the elastic extensional matrix, [D] is the flexural rigidity matrix, and
[B] is the extension coupling matrix of the laminated plate.
Element Displacements, Matrices and Equations
Proceeding from this point, the displacements in equation (1) are approximated
over a typical plate element , e.g. rectangular [8] or triangular [9], using the
corresponding interpolation functions. The in-plane displacements and the linear
strains are interpolated from nodal values by
1376
where [Hm] and [Bm] denote the displacement and strain interpolation matrices,
respectively, and {Wm} is the in-plane nodal displacement vector. The transverse
displacement, slopes and curvatures are interpolated from the nodal values by
w = [Hb]{w^}, |^’^| = [G]{wb}, {K} = [B|,]{wb} (4a,b,c)
where [HJ and [G] and [BJ denote the bending displacement, slope and curvature
interpolation matrices, respectively, and {wb } denotes the nodal transverse
displacements and its derivatives. Through the use of Hamilton’s principle, the
equations of motion for a plate element undergoing large amplitude vibration may
be written in the form
[hb]
0
T [H]
[kB]l
r[klNm]+[klNB] [klbml
,r[k2b] <
[k»]J
L ^'"*1 0 J'
[ 0 (
or
[m]{ w} + m + [kl] + [k2] }{w} = 0 (5)
where [m] and [k] are constant matrices representing the element mass and linear
stiffiaess characteristics, respectively; Pel] and Pe2] are the first order and second
order non-linear sfiffiiess matrices, respectively; PcInJ depends linearly on
unknown membrane displacement ({Nm }= [A][Bni]{wni}); PcInb] depends linearly
on the unknown transverse displacement ({NB}=[B][Bb]{Wb}); DelbnJ depends
linearly on the unknown plate slopes and represents coupling between membrane
and bending displacements; and [k2b] depends quadratically on the unknown plate
slopes.
System Equations
After assembling the individual finite elements for the complete plate and
applying the kinematic boundary conditions, the finite element system equations of
motion for the large-amplitude free vibration of a thin laminated composite plate
can be expressed as
[M]{w}4- ([K]+ [K1(W)] + [K2(W)]){W} = 0 (6)
where [M] and [K] are constant matrices and represent the system mass and
stifiBness respectively; and [Kl] and [K2] are the first and second order nonlinear
stifi&iess matrices and depend linearly and quadratically on the unknown structural
1377
nodal displacements { W}, respectively. Most of the finite element large amplitude
fi-ee vibration results for plates and beams in the literature, e.g. references [1,6] and
others, were based on eq. (6) using an iterative scheme and various approximate
procedures. The system equations are not suitable for direct numerical integration
because: a) the nonlinear stif&iess matrices [Kl] and [K2] are functions of the
unknown nodal displacements, and b) the number of degrees of fireedom (DOF) of
the system nodal displacements {W} is usually too large. Therefore, eq. (6) has to
be transformed into modal or generalized coordinates followed by a reduction of
the number of DOF. In addition, the general DufiSng-type modal equations will
have constant nonlinear modal stifihiess matrices. This is accomplished by a modal
transformation and truncation
r=l
where and cou are the natural mode (normalized with the maximum
component to unity) and linear firequency from the eigen-solution ©|_j.[M]{
(|))«=[K]{ (|)}®.
The nonlinear stififtiess matrices [Kl] and [K2] in eq. (6) can now be expressed
as the sum of the products of modal coordinates and nonlinear modal stiffaess
matrices as
[Kl] = ^q,(t)[Kl((])«)] (8)
r=I
n n
[K2] = ^ ^ qr (t)qs (t)[K2((|)('') , )] (9)
r=l s=l
The nonlinear modal stifihiess matrices [Kl]^"^ and [O]^'^^ are assembled from the
element nonlinear modal stif&iess terms [kl]^^^ and Pc2]^“^ as
([K1]«.[K2]‘“))= ^([klf>,[k2]'“>) (10)
al! elements
+ bdy. conds.
where the element nonlinear modal stif&iess matrices are evaluated with the known
linear mode Thus, the nonlinear modal stiffiiess [Kl]^^^ and [K2]^'®^ are constant
matrices. Equation (6) is thus transformed to the general Duf6ng-type modal
equations as
1378
(11)
[M]{q}+([K]+[KlJ+[K2„]){q} = 0
where the modal mass and linear stifhiess matrices are diagonal
([M],[K]) = [<I.f([M],[K])[<I.]
and the quadratic and cubic terms are
LI
[Kl,]{q} = [<Df J^qjKl]
1^)
Vt=1
n n
(rs)
r=:l s=l
\Wk}
(12)
(13)
(14)
AH modal matrices in eq. (11) are constant matrices. With given initial
conditions, the modal coordinate responses {q} can be determined from eq. (11)
using any direct numerical integration scheme such as the Runge-Kutta or
Newmark-P method. Therefore, no updating of the vibration modes is needed [10].
For periodic plate osciHations have the same period T, the response of all modal
coordinates should also have the same period T. Since the initial conditions wHl
affect greatly the modal response, the determination of initial conditions for periodic
plate osciHations is to relate each of the rest modal coordinates in powers of the
dominated coordinate as
arqi(t;IC) + brqi(t;IC) + Crq^(t;IC)+ . = qr(t;IC), r = 2,3. ...n
(15)
where the 2k, br, Cr , . are constants to be determined, and IC denotes initial
conditions. For a three-mode (n=3) system, it is accurate enough to keep up to the
cubic term only in eq. (15) and this leads to two set of equations
a2qi(tp;A,B,C) + b2qf(tp;A,B,C) + C2qJ(tp;A,B,C) = q2(tp;A,B,C), p = 1,2,3
a3qi (tp ; A, B, C) + bsq J (t p ; A, B. C) + Csq J (tp ; A. B, C) = q3 (tp ; A, B, C), p = 1,2,3
(16a,b)
in which the modal coordinates qi, q2 and qs at tp are known quantities and the
initial conditions are qi(0)=A, q2(0) =B, q3(0)=C and qi(0) = q2(0) = q3(0) = 0.
PracticaHy, only eight equations are needed to determine the eight unknowns a2, as.
1379
hi, b3, C2, C3, B and C through an iterative scheme. However, the number of
equations can be more than the number of unknowns for accurate determination of
initial conditions and the least square method is employed in this case.
The time history of the plate maximum deflection can be obtained from eq. (7).
The participation value from the r th linear mode to the total deflection is defined
as
maxjqj.
n
^max|qi
i=l
(17)
Thus, the minimum number of the linear modes for an accurate and converged
frequency solution can be determined based on the modal participation values.
Results and Discussions
Assessment of Single-Mode Elliptic Function Solution
The fundamental frequency ratio Co/col = 2.3051 at Wmax/r =5.0 for a simply
supported beam obtained by Woinowsky-Kiieger [2] using a single-mode and
elliptic fimetion is assessed first. The conventional beam element having six (four
bending and two axial) DOF is used. A half-beam is modeled with 10, 15, 20
elements, and the lowest four symmetrical linear modes are used in the Duffing
modal equations. Table 1 shows that a 20-element and 1-mode model will yield a
converged result. The percentages of participation from each mode for various
values of Wmax/r are given in Table 2. The modal participation values demonstrate
that a single mode (n=l) will yield an accurate fundamental frequency because the
contribution from higher linear modes to the total deflection is negligible (< 0.01 %
for Wmax/r up to 5.0). There is a small difference in frequency ratios between the
present finite element and the elliptic integral solutions. This is due to the difference
between the axial forces of the two approaches, the finite element method (FEM)
gives a non-uniform axial force in each element; however, the average value of the
axial force for each element is the same as the one in the classic continuum
approach. The lowest (2.0310) and the highest (3.3438) frequency ratios at Wmax/r
=5.0 in reference [5] are not accurate.
Frequency ratios for higher modes of the simply supported beam are obtained
next. A model with 40-elements and 3-anti-symmetric modes for the whole beam is
employed for the frequency ratio of the second nonlinear mode. The mode
participations shown in Table 2 indicate that a single-mode approach will yield
accurate frequency results. And the frequency ratios for the second mode are the
same as those of the fundamental one. Thus, the present method agrees extremely
well with Woinowsky-Krieger’s classic single-mode approach.
1380
The time history of the first two symmetric modal coordinates and the beam
central displacement, phase plot, and power spectral density (PSD) at maximum
beam deflection W^Jr = 5.0 for the fimdamental firequency (or mode) are shown in
Fig. 1. The time scale is non-dimensional and Ti is the period of the fundamental
linear resonance. It is noted that although the central displacement response looks
like a simple harmonic motion, it does have a small deviation fi:om pure harmonic
motion due to the second small peak in the spectrum. This is in agreement with
classical solution that the ratio of the jfrequency of the second small peak to that of
the first dominant peak is 3.
Now we are ready to assess the single-mode fundamental firequency of a simply
supported square plate obtained by Chu and Herrmann [3]. A quarter of the plate is
modeled with 6 x 6, 7 x 7, 8 x 8 and 9 x 9 mesh sizes and 1, 2, 4 or 5 symmetrical
modes are used. The conforming rectangular plate element with 24 (16 bending
and 8 membrane ) DOF is used. The in-plane boundary conditions are u = v = 0 on
all four edges. Table 3 shows that the 8 x 8 mesh size in a quarter-plate and 4-
mode model should be used for a converged and accurate frequency solution. Table
4 shows the fi:equency ratios and modal participation values for the lowest three
modes at various Wmax/h for a simply supported square plate (8x8 mesh size in a
quarter-plate). It indicated that at least two linear modes are needed for an accurate
frequency prediction at Wmax/h =1.0, and the contribution of higher linear modes
increase with the increase of plate deflections. The modal participation values also
show that the combined modes (1,3)-(3,1) and (2,4)-(4,2) are independent of the
large-amplitude vibrations dominated by (1,1) and (2,2) modes, respectively. The
time history, phase plot, and PSD at the maximum deflection Wmax/h =1.0 for the
fundamental mode are shown in Fig. 2a, and Tu is the period of the fundamental
linear resonance. There is one small peak in the spectrum and the frequency ratio of
the second small peak to the first dominant one is 3. The low (1.2967) and the high
(1.5314) frequency ratios at Wmax/h =1.0 given in reference [6] are not accurate.
The influence of the initial conditions on periodic motion is demonstrated in Fig.
2a and 2b. In Fig. 2a, the modal coordinates all have the same period, and the initial
conditions are determined fi*om eq.(15). They are qii(0)/h=1.0, qi3+3i(0)/h= -
0.0155, qi3.3i (0)/h = 0.0, q33(0)/h=0.000813, and qi5+5i(0)/h= 0.00011, and initial
velocities are null, whereas in Fig. 2b, qii(0)/h=1.0 and all others are nuU. The
modal coordinates do not have the same period.
Clamped Beam
It is thus curious to find out whether multiple-mode is required for the clamped
beam. Convergence study of the fundamental firequency ratios at Wmax/r =3.0 and
5.0 shown in Table 5 indicates that a 25-element (half-beam) and 4-mode model
win yield accurate and converged results. The time history, phase plot and PSD at
Wmax/r =5.0 are shown in Fig. 3. The modal participation values in Table 6 and the
PSD in Fig. 3 confirm that at least two modes are needed for accurate firequency
results.
1381
Symmetric Composite Plate
A simply supported eight-layer symmetrically laminated (0/45/-45/90)s
composite plate with an aspect ratio of 2 is investigated. The graphite/epoxy
material properties are as follows; Young’s moduli Ei = 155 GPa, E2 = 8.07 GPa,
shear modulus Gn = 4.55GPa, Poisson’s ratio V12 =0.22, and mass density p =
1550 kg/m^ A 12 X 12 mesh is used to model the plate. The in-plane boundary
conditions are fixed (u=v=0) at all four edges. The first seven linear modes are used
as the modal coordinates. Table 7 gives the fundamental firequency ratios and mode
participation values for the linear modes in increasing firequency order. The modal
participation values indicate clearly that four modes are needed in predicting the
nonlinear fi-equency, and other three of the seven are independent of the
fundamental nonlinear mode. Figure 4 shows the time-history, phase plot, and PSD
at Wmax/h =1.0.
UNS YMMETRIC COMPOSITE PLATE
A simply supported two-layer laminated (0/90) composite plate of 15 x 12 x
0.048 in. (38 x 30 x 0.12 cm) is investigated. The graphite/epoxy material
properties are the same as those of the symmetric composite plate. A 12 x 12 mesh
is used to model the plate. The in-plane boundary conditions are fixed at all four
edges. The first four linear modes are used as the modal coordinates. Table 8 gives
the fundamental firequency ratios and mode participation values for the linear modes
in increasing fi-equency order. From the phase plot, the time histories and PSD
shown in Fig. 5, it can be seen that the total displacement response has a non-zero
mean (i.e. the positive and negative displacement amplitudes for all modal
coordinates are not equal). The quasi-ellipse in the phase plot is not symmetrical
about the vertical velocity-axis. In the PSD at Wmax/h =1.0, it is observed that there
are four small peaks in the spectrum and the firequency ratios of the second, third,
fourth and fifth peak to the first dominant one are 2, 3, 4 and 5, respectively. This
observation indicates that the displacement response includes the
superharmonances of orders 2, 3, 4, and 5. The curves, which the positive and
negative displacement amplitudes are plotted against the fundamental firequency
ratio, are also given in Fig. 5. The difference between the positive and negative
amplitudes increases as the firequency ratio increasing.
Conclusions
A multimode time-domain formulation, based on the finite element method, is
presented for nonlinear firee vibration of composite plates. The use of FEM enables
the present formulation to deal with composite plates of complex geometries and
boundary conditions, and the use of the modal coordinate transformation enables to
reduce the number of ordinary nonlinear differential modal equations to a much
smaller one. The present procedure is able to obtain the general Duffing-type modal
equations easily. Initial conditions for all modal coordinates having the same time
1382
period are presented. The participation value of the linear mode to the nonlinear
deflection is quantified ; they can clearly determine the minimum number of linear
modes needed for accurate nonlinear frequency results.
The present fundamental nonlinear fi-equency ratios have been compared with
the single-mode solution obtained by Woinowsky-Kneger for simply supported
beams and by Chu and Herrmann for simply supported square plates. The
Woinowsky-Krieger’s single-mode solution is accurate. For all other solutions,
however, two or more modes are needed. The nonlinear firequencies for
symmetrically and unsymmetrically laminated rectangular composite plates are also
obtained. The phase plot and power spectral density showed that nonlinear
displacement responses are no longer harmonic, and multiple modes are required
for isotropic clamped beams and isotropic and composite plates.
References
1. M. Sathyamoorthy 1987 Applied Mechanics Review 40, 1553-1561. Nonlinear
vibration analysis of plates: A review and survey of current developments.
2. S. Woinowsky-Kreger 1950 Journal of Applied Mechanics 17, 35-36. The effect of an
axial force on the vibration of hinged bars.
3. H. N. Chu and G. Herrmann 1956 Journal of Applied Mechanics 23, 523-540.
Influence of large amplitudes on jfree flexural vibrations of rectangular elastic plates.
4. G. Singh, A. k. Sharma and G. V. Rao 1990 Journal of Sound and Vibration 142, 77-
85. Large amplitude free vibration of beams-discussion of various formulations and
assumptions.
5. H. R. Srirangaraja 1994 Journal of Sound and Vibration 175, 425-427. Nonlinear free
vibrations of uniform beams.
6. S. R. Rao, A. H. Sheikh and M. Mukhopadhyay 1993 Journal of the Acoustical
Society of America 93 (6), 3250-3257. Large-amplitude finite element flexural
vibration of plates/stiffened plates.
7. Y. Shi and C. Mei 1996 Journal Sound and Vibration 193, 453-464. A finite element
time domain modal formulation for large amplitude free vibration of beams and plates.
8. K. Bogna:, R. L. Fox and L. A. Schmit 1966 Proceeding of Conference on Matrix
Methods in Structural Mechanics, AFFDL-TR-66-80, Wright-Patterson Air Force
Base, Ohio, October 1965, 397-444. The gena-ation of interelement compatible
stiffness and mass matrix using the interpolation formulas.
9. Teseller and T. J. R. Hughes 1985 Computer Methods in Applied Mechanics and
Engineering 50, 71 -101 . A three node Mindlin plate element with improved fransverse
shear.
10. A. K. Noor 1981 Composites and Structures, 13, 31-44. Recent advances in reduction
methods for nonlinear problems.
1383
Table 1 . Convergence of the fundamental frequency ratio at Wmax/r =5.0 for a
simply supported beam
No. of elements
and 4 modes
(C0/(0l)i
No. of modes
and 20 elements
(G)/G)l)i
10
2.3537
1
2.3506
15
2.3511
2
2.3506
20
2.3506
3
2.3506
--
4
2.3506
Table 2 The lowest two frequency ratios and the modal participations for a
simply supported beam
W^nax/r
Elliptic
integral [2]
(CO/OOl)!
FEM
Modal
Participation %
(CO/COl)!
_ _
0.2
1.0038
1.0038
100.00
0.000
0.000
0.4
1.0150
1.0149
100.00
0.000
0.000
0.6
1.0331
1.0331
100.00
0.000
0.000
0.8
1.0580
1.0581
100.00
0.000
0.000
1
1.0892
1.0892
100.00
0.000
0.000
2
1.3178
1.3179
100.00
0.002
0.000
3
1.6257
1.6258
100.00
0.004
0.000
4
1.9760
1.9761
99.99
0.005
0.000
5
2.3501
2.3506
99.99
0.009
0.000
W.„ax/r
(co/g)l)2
(C0/CDl)2
q.2_ _
q4
Q6
0.2
1.0038
1.0038
100.00
0.000
0.000
0.4
1.0150
1.0149
100.00
0.000
0.000
0.6
1.0331
1.0332
100.00
0.000
0.001
0.8
1.0580
1.0582
100.00
0.000
0.001
1
1.0892
1.0893
100.00
0.000
0.002
2
1.3178
1.3181
99.99
0.000
0.006
3
1.6257
1.6260
99.98
0.000
0.015
4
1.9760
1.9768
99.98
0.000
0.021
5
2.3501
2.3512
99.96
0.000
0.037
Table 3. Convergence of the fundamental frequency ratios for a simply
supported square plate (Poisson’s ratio=0.3)
Mesh sizes
and
(CD/Qjn
at
W^,Jh
No. of modes
and
(co/coUn
at
W^Jh
4 modes
1.0
1.4
8x8 mesh
1.0
1.4
6x6
1.4174
1.7423
1
1.7028
7x7
1.4163
1.7396
2
1.4169
1,7433
8x8
1.4164
1.7403
4
1.4164
1.7403
9x9
1.4164
1.7400
5
1.4163
1.7401
Table 4. The lowest three frequency ratios and the modal participations for a
simply supported square plate (Poisson’s ratio=0.3)
Elliptic
FEM
integral [3]
(CO/ffiiJii
(to/oiOii
Modal
Participation
%
qu
qi3 -1- qsi
qi3 - qsi
q33
qi5 -f. qsi
0.2
1.0195
1.0195
99.93
0.07
0.00
0.00
0.00
0.4
1.0757
1.0765
99.72
0.27
0.00
0.01
0.00
0.6
1.1625
1.1658
99.38
0.59
0.00
0.02
0.00
0.8
1.2734
1.2796
98.93
1.02
0.00
0.05
0.01
1.0
1.4024
1.4163
98.34
1.57
0.00
0.08
0.01
1.2
1.5448
1.5659
97.54
2.30
0.00
0.15
0.01
1.4
1,6933
1.7401
96.29
3.42
0.00
0.27
0.02
(C0/®l)21
q2i
q23
q4i
q43
...
0.2
N/A
1.0243
99.93
0.06
0.01
0.00
...
0.4
N/A
1.0976
99.50
0.45
0.03
0.01
—
0.6
N/A
1.2072
98.15
1.28
0.54
0.02
...
0.8
N/A
1.3411
97.54
2.41
0.00
0.05
—
1.0
N/A
1.5126
96.24
3.69
0.00
0.08
...
1.2
N/A
1.6900
94.90
4.92
0.03
0.15
...
1.4
N/A
1.8952
93.54
6.01
0.01
0.44
...
y/raJh
(C0/0)l)22
q22
q24 + q42
q24 - q42
q44
...
0.2
N/A
1.0245
100.00
0.00
0.00
...
0.4
N/A
1.0751
100.00
0.00
0.00
0.00
...
0.6
N/A
1.1611
99.99
0.00
0.00
0.01
...
0.8
N/A
1.2806
99.98
0.01
0.00
0.01
...
1.0
N/A
1.4041
99.93
0.01
0.00
0.06
—
1.2
N/A
1.5551
99.97
0.01
0.00
0.01
...
1.4
N/A
1.7074
99.98
0.02
0.00
0.00
—
Table 5, Convergence of the fundamental frequency ratios for a clamped beam
No. of
elements
and 4 modes
(co/©l)i
at
W™/r
No. of modes
and
(©/©lOi
at
Wn.ax/r
3.0
5.0
25 elements
3.0
5.0
10
1.1751
1.4046
1
15
1.1740
1.4009
2
20
1.1732
1.3999
3
25
1.1731
1.3996
4
1385
Table 6 The fundamental frequency ratios and the modal participations for a
clamped beam
Elliptic
integral
(0)/(0l)i
FEM
(ffl/(0L)l
Modal Participation
%
qi
q3
qs
_
1.0222
1.0222
99.78
0.20
0.02
0.00
2.0
1.0857
1.0841
99.33
0.58
0.08
0.02
3.0
1.1831
1.1731
98.35
1.44
0.17
0.04
4.0
1.3064
1.2817
97.37
2.28
0.29
0.07
5.0
1.4488
1.3996
96.26
3.22
0.42
0.11
Table 7 The fundamental frequency ratios and the modal participations for a
simply supported rectangular (0/45/-45/90)s composite plate (a/b=2)
WmJh
Modal
Participation
%
(ffl/coOn
qu
qi2
q2i
qi3
q22
q23
qsi
0.2
1.0408
99.51
0.00
0.00
0.41
0.07
0.00
0.02
0.4
1.1490
96.57
0.00
0.00
3.01
0.24
0.00
0.17
0.6
1.3484
92.93
0.00
0.00
4.55
0.47
0.00
2.04
0.8
1.5241
98.51
0.00
0.00
0.53
0.94
0.00
0.02
1.0
1.7190
97.43
0.00
0.00
2.39
0.11
0.00
0.07
1.2
1.9258
95.78
0.00
0.00
3.57
0.62
0.00
0.02
1.4
2.1409
94.27
0.00
0.00
4.84
0.77
0.00
0.13
Table 8
The fundamental frequency ratios and the modal participations for a
simply supported rectangular (0/90) composite plate
W„,Jh
Modal Participation
%
(CO/COiOn
qii
qi3
qsi
q33
0.2
1.0358
97.83
1.18
0.82
0.18
0.4
1.1432
95.13
2.25
2.24
0.38
0.6
1.2993
94.53
3.86
1.18
0.60
0.8
1.5432
88.56
4.36
4.77
2.31
1.0
1.7880
89.15
3.31
5.06
2.48
1.2
2.0142
92.22
2.89
3.15
1.74
1.4
2.2823
92.01
2.92
2.92
2.15
1386
0.5 1 1.5 2 -6 -3
Time Ratio (tn‘1)
4 6 8 10 12
Frequency Ratio
U
Displacement
Figure 1. Time histories, phase plot and PSD for the fundamental
mode at =5.0 of a simply supported beam
Power Spectrum Density Total OlsplacemonllThIckness qSSn'hJcknBSS q1 1/ThIckness
p P P T* Thousandths o o o
Time Ratio (tn"11)
-0.4 -0.2 0 0.2 0.4
Displacement
Figure 2a. Time histories, phase plot and PSD for the fundamental
mode at W^u/h =1.0 of a simply supported square plate
1388
Thousandths
Displacement Spectrum Density Total Displacement/Thickness q22/Thickness q1 IH'hickrtess
0 12 3 4
Time Ratio {t/T11)
0 12 3 4
0 2 4 6 8 10 12
Frequency Ratio
0 12 3 4
Time Ratio {tfTII)
0 12 3 4
Time Ratio (t/TII)
Displacement
Figure 4. Time histories, plot and PSD for the fundamental
mode at Wmax/h =1.0 of a simply supported
(0/45M5/90)s rectangular plate
1391
Power Specirum DonsUy Total Dlsplacemenin'hlckness qSin’hlckness q11/Thlckness
NONLINEAR FORCED VIBRATION OF BEAMS BY THE
HIERARCHICAL FINITE ELEMENT METHOD
P. Ribeiro and M. Petyt
Institute of Sound and Vibration Research, University of Southampton, Southampton
S017 IBJ, UK
Abstract: The hierarchical finite element (HFEM) and harmonic balance
methods are 
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